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One thing that's generally been true as long as races have been set up with bonuses to certain stats (and sometimes penalties to others) is that the stats have had a stronger "pull" than the rest of everything else that's going on, with relatively few exceptions. (Though some certainly exist.)

Imagine two setups. I'm going to phrase them in extremes:

I) A Half-Orc is a great fighter. A Half-Elf is an okay fighter. A Halfling is a bad fighter.
II) A Half-Orc, a Half-Elf and a Halfling are all good fighters, but have bonuses that help them be a fighter in ways that are culturally appropriate. For example, a half-orc might have a bonus to charging, a half-elf might give bonuses to allies, and a halfling might be effective vs. larger characters.

Obviously D&D has never been completely at one extreme or the other, but I think that moving more towards II is generally more interesting. It's even okay if most people would agree that a Half-Orc is a better fighter than a Halfling, but a Halfling fighter still has unique and awesome options available that make playing one something more than just a pure RP choice at the cost of adventuring utility.

I hope that most people like that setup even if they disagree with what I suggest next.

I think that a great way to move hard towards setup II is to chuck racial ability score bonuses and replace them with more potent general racial features. While we're at it, it would be cool if every race got at least one feature for each of the three pillars. (For combat, defensive abilities are better than offensive ones because every character benefits from them.)

The big drawback to this is that it makes races more complicated. An ability score boost goes on the sheet and then you can forget about it. As someone who doesn't like a proliferation of tiny situational bonuses, I'm leery of making race a bigger source of them. But I like the idea that races have bonuses to being a fighter differently than other races instead of more or less just better or worse than other races.

Also this idea might just be terrible.

I think removing ability bonuses sounds great.

For example, to show that dwarves are naturally tough and hardy, say they get to roll their skill die with Constitution ability checks instead of giving them a Con bonus.

If they get rid of racial bonuses, what they should do is create Weapon Bonuses and other benefits (like what the other races receive) that can be in a pool for Human PCs.   Then let each Human PC pick two or three of them.   That would simulate the diversity that is represented in with Humans in the fantasy world, and it would balance them with the other races.   I'm all for getting rid of the bonuses to abilities.   They are unnecesary because a) they aren't interesting, b) they don't really fit with bounded accuracy game mechanics.  

I think the OP has a point. Stat bonuses may have a place, but they tend to force players to choose between "the character I wanna play" and "the obvious powergaming choice". To me, that's an unfortunate and unnecessary situation

giving the races general bonuses that actively influence how the character solves problems etc should be the way to go.  

Example: both halflings and half-orcs can pick up the rogue class and become good sneaks. While the halflings size gives him advantages at darting between the legs of his enemies and planting a blade in their knee, the half-orc's size lets him move fast and force his way into an advantageous position, or maybe subdue his foes bluntly before striking with a well-placed dagger.

hardcoded racial bonuses are boring and restrictive