What is D&D? A miserable pile of sacred cows.
Let me start by saying that I am not an edition warrior. I have some preferences for 4E and Pathfinder, but I largely see many of the same issues in all of them. Rather, I come from an outside perspective as someone who would prefer that WotC make an objectively better game, not simply please some groups which may or may not be pleasable. To that end, I'm going to point out what I see as serious CORE flaws, ones that hold the entire edition back, and that can't simply be fixed by expecting everyone to buy splat books called "modules".
So let's start from the absolute basics, attributes. We've had six ability scores for a while now, and the basic premise is fine, in that it attempts to give a triad of physical and mental stats which should altogether cover pretty much anything. That's good. What isn't, however, is the lack of balance, and sometimes lack of definition in a few of them.
So first off, Strength. The only non-skill or attack based function it provides is carrying capacity, which historically becomes moot due to things like bags of holding. It typically helps with things like jumping, swimming, and climbing, but at the same time these activities are often penalized by armour to begin with, and strong characters tend to be the armour wearing types, so this evens it out, if anything. Then we have the combat aspect, it only acts as a bonus to attacks and damage. What this means, is we have a stat that effectively is only useful for it's attack bonus, because the damage bonus is small and will never scale to match monster HP, but the attack bonus will always be useful. And on top of that, attack bonuses like this destroy the concept of bounded accuracy, which in turn means that this stat either keeps that effect and breaks the purpose of bounded accuracy, or loses it and effectively governs nothing. Either way, it needs work.
Next, Constitution. This one has been an age old problem, because a stat that governs HP will automatically be useful to everyone, but more importantly it simply controls too much of player HP. If we look at the average HP per level, Wizards and Rogues currently get 4, Clerics get 5, and Fighters get 6. What this means, is a 14 Con is about a 50% HP boost to the lower health classes, and 16 Con is a 50% health boost to a Fighter. The only stat somewhat comparable in how much of an effect it gives is Dexterity, which I'll get to in a moment. Constitution also governs saves against quite a few effects, many of them lethal if a save is failed, once again making it possibly too important.
Dexterity is thought of by some to be a "super stat", and honestly it's true. Governing attack and damage of finesse and ranged weapons, AC, initiative, several types of checks, as well as still governing the same saving throws as reflex, it's doing far too much. In particular, as noted with Strength, the attack part would have to go, while the damage doesn't scale and has the same issue of starting powerful and then becoming unimportant. The AC boost could also be argued to be a bad thing for bounded accuracy, since in that kind of system it's important for monsters to be able to hit the players as well. And as for initiative, it's arguable, because the swingier the game is, the more imbalancing it is, because winning initiative can mean winning the fight.
Intelligence, Wisdom and Charisma all share very similar problems. Simply put, they have nothing to properly govern as far as stats are concerned, acting only as stats for skill checks, savings throws, and govern the attack of a magic user, which as noted more than once, is a bad idea for bounded accuracy. While their influence on skill checks are fine, they lack any tangible way to affect combat, their saving throws are ambiguous due to being split off from Will saves, and in the case of Wisdom and Charisma, the stats themselves are difficult to properly define. Wisdom apparently reflects perception, insight, and... religion for some reason? If it weren't for the random tying of this stat to divine magic, you might easily be able to simply call it Perception, as that is what it otherwise governs and seems to make an awful lot more sense.
Then there's Charisma, which suffers from a few problems. First off, it's the "social stat" making it's usefulness entirely dependent on both the DM creating appropriate social situations, and the players using social checks rather than trying to fight or avoid interaction. What this also causes though, is a discrepancy between roleplaying and checks, where high Charisma can allow a player to be accomplished in social interaction without actually roleplaying, and conversely a good roleplayer can be penalized for not putting enough points into Charisma to ensure success. There is also of course the issue of immense overlap between Wisdom saving throws and Charisma saving throws, to the point where the example saves for Wisdom(charms, fears, being influenced) really make more sense for Charisma, as in this case it helps protect against anything that mess with sense of self. The last thing I have to say on Charisma, is as it represents force of personality, sheer force of will, and the ability to influence others, perhaps it ought to be known as Willpower instead? If there's an issue of influencing others not always making sense, with the way checks and skills work it hardly matters, just pick the score that DOES fit for that situation.
Next, your four "core" classes. Along with ability scores needing work, if you are to base all future classes off of your first four, you're going to need a very strong foundation, otherwise the game will fall apart. Due to the nature of the classes, and their interaction being the important part, I won't cover them individually, but I will instead compare them and comment on the whole.
So the first issue, is you need to decide what core four really means. Is it playstyle, role, or just theme? Because as it is right now, it seems more like theme than anything, because while the four are different, it's not in a way that's good for the game. Three pillars of the game have been brought up before, and that applies here. Every class needs to be able to contribute equally in combat, exploration, and social situations. Social situations above all others is more RP and DM based, so class influence in that area should be fairly slim, but still equivalent, leaving that area of progression to the story and interaction. That leaves combat, and exploration, which absolutely MUST be equal. Note, that equal doesn't mean that what a class provides needs to be the exact same as another, only that the output is even.
So, we have a class that by name alone claims that it must be the best at fighting, which is already a bad idea. We have a class that defines itself by only using skills, acting as an opposite almost, which is also bad because this means that the Rogue will always outperform the Fighter in exploration and social areas, while failing at combat, when really the two should be even with different approaches. We then have a class that arguably has too much influence on the flow of the game due to a monopoly on healing, while still being generally effective in the other areas. And lastly we have a class that seems to be specifically designed as a one round winner type of class, where they simply steal the spotlight from the rest of the players by expending a limited resource to simply end fights outright much of the time, using that same ability to generally instantly succeed and exceed what the other classes can do in exploration as well. All in all, it's a mess, the Fighter needs more exploration and out of combat abilities, the Rogue needs less instant success on every skill ever and better but different combat ability than the Fighter, the Cleric can't be the only guy with healing, and needs to avoid out of hand spells, but is otherwise a decent base gish, and the Wizard needs to stop doing everyone's job better, because a limited number of uses doesn't make that balanced.
If you need a brainstorming starting point, consider the 4E roles, but don't stop at combat. The four core classes already largely comprise of those combat roles, but define four different ways of doing exploration, and possibly four different ways of handling social situations. You can consider these to be similar to themes/specialties/backgrounds, and as a result you open up the possibility of mixing and matching combat style and exploration style to give different approaches to classes.
Oh, and quit it with the dumb Fighter needs to be simple thing. Just give every class basic options(That means Wizards and Clerics too), and then build upon them with choices for every class to make. This means let every class have a bunch of things they could choose to do every round WITHOUT making basic attacking objectively better, which unfortunately is how the Fighter currently is. Also fill up dead levels and give options as characters level. if you only want to give us ten levels base because making anything higher is too "hard" for you, then at the very least make them meaningful. there is literally zero point in just getting higher numbers and stronger versions of the same things other than to please people who like bigger numbers. I'm not saying do away with leveling altogether, but more choices should be a core feature, not a module. The last note on classes I have is, first step to balancing out casters and avoiding five minute workdays or one spell encounter enders, is do away with vancian casting as core, just saying.
Lastly, races. I don't have much to say here, because it's all pretty much been said, but what I will say is this. Give humans a real feature, endurance or ingenuity could be a starting point, but make it something that actually effects how they play, make it something that isn't objectively superior to other races, and stop it with the versatility thing, that's most of the problem with your Wizard design too. Separate racial and cultural benefits into two neat packages, dwarves may always be tough, but they might not always use hammers, for example. On a similar note, make the benefits of the races useful to any class. This is something you tried to do somewhat in 4E, but you seem to have completely abandoned it at this point. This means stop giving them weapon based bonuses as a major trait, it pigeon holes characters into picking those options, and it doesn't benefit any character that can't or wouldn't use those weapons.
I agree with a lot of this. When backgrounds were announced, I was so excited - they could now make all classes contribute only to combat, and backgrounds contribute only to out of combat. In that model, a rogue is a "mobile warrior/acrobat", and a wizard is a "spell caster/spell caster." That allows clerics, who tend to lack out of combat options, to be focused on "spell caster generalist" or face or guru, depending on what the player wants.
However, the feedback on the forums tends to be that tradition is important, whether it makes the game good or not. While I understand the appeal of tradition, I buy games on their merits, not on their nostalgia.
I just feel like the developers have blind spots. The whole discussion of magic users and different casting systems _never_ got to the concept of easy to run spell users. It was always the spell casters are the classes the experts play, where the new players should play fighters. I just don't understand why we rely on tying archetypes to play styles - if I'm new and want to play Harry potter, why do I have to play Conan, just because its easier? Why not have a easy wizard?