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This is basically how I play 4E. 

Still I would prefer if 5E classes had the 'ability' to perform various roles without being forced to do only that.

In principle I think that's exactly what most folks want. 3e actually does this but it doesn't make you come out and say "I'm gonna be a tank" on your character sheet. The tank feats and spells are there; the player is free to pick them.

One of the things I like about the 5e we've got so far is they allow us to pick those roles but they don't force a class to always be one role (exactly as you're describing). A fighter can pick the fighting style to fill a specific range of roles in the group - mostly tank or striker but there's some variety in each. Clerics are all over the shop as well with domains basically determining all of their baseline proficiencies and giving spells that push them into drastically different sorts of play styles.

Rogues and wizards, I feel, have less flexibility going for them right now but maybe that'll change.

Just using 3e as an example (might be a bad idea... we'll see), the only role a wizard can't really fill is a healer. I know there were mechanical flaws and in practise the wizard wasn't as flexible but that wasn't the fault of the spells so much as it was the way monster saves scaled up. A 3e rogue isn't as flexible in practise as it was in theory either but there was potential for them to fill a wide variety of roles in the party.

I think my main complaint about roles, aside from the mistake of forcing classes to only perform one function, is that it creates a situation where people perceive that their class is only good in combat. I know the roleplaying doesn't have to depend on the mechanics but the way the mechanics are presented has an impact on how people roleplay, for better or worse.

Well, it's something at least.  We wouldn't need to come up with a whole new "Avenger" class if a normal paladin could just choose to be either a tank or a healer or a striker.  On those grounds, the idea has merit.

It really doesn't "solve" the problem of roles, though.  You still make one decision (to be a tank or a healer or a striker or a controller) and that heavily influences every action you'll ever take for the duration of the campaign.  I hate to bring up the WoW comparison (is that like the new Godwin's Law around here?), but it is really directly analogous to how they deal with their class system over there: each class has three sub-classes, and each sub-class has one specific role, and if you try to violate that role then you are "doing it wrong".

My opinion has always been that the choices you make in combat should be "what to do" and not "how to do it"; rather than choosing between a big heal or a small heal, or choosing between a heavy strike or two light strikes, or choosing whether to entangle a foe or to blind a foe, the question should be whether to heal or to strike or to control.

Acting within a role does not make for interesting choices - there is almost always a "best" answer.  Choosing which role to act upon does make every choice interesting, because the options are not directly comparable.

Yes, 5E so far is quite good in this respect.
The only thing I am a bit worried about is that all classes seem to need its own 'system'. Every class uses its own rules and its own resources. I think this treathens to limit players options to compose the character they want (and get the role they want), and it will complicate multiclassing.

That some classes had their own system (Wiz, Sor, Warlock) to do similar things was one of the annoying parts of 3e..
I hope some mechanical overlap will appear when new classes are added to the playtest.. 

If the ranger is driven by Thrill of the Hunt points and the Warlord has a number of shouts per day and level... then things are going in the wrong direction.... =/