But on the other side, I think that ability scores ought to be relatively modest at start (showing raw talent) and then improved only in relation to what the characters are. That is, a melee fighter should get occasional boosts to Str...
Actually, I would go the other way - a character's ability scores are generated at character creation, they are
maybe modified by race and initial class choice, and after that they don't change. The ability scores represent the character's aptitudes for various things, and don't vary with time - everything else comes with boosting attack bonuses, skills, and so on when levelling.
Under this model, Batman didn't gain in Str, Dex etc through all that training - he gained lots of levels which boosted his many skills. (Plus, he rolled really well as a starting character.
I would also reduce eliminate ability drain and damage (as 4e has done), any item/spell/effect that increases or decreases stats, and even ability checks - everything that was an ability check now becomes a skill check of some sort.
Finally, I would reduce the magnitude of bonuses for ability scores: -3 at 3 or less, -2 at 4-6, -1 at 7-9, 0 at 10-11, +1 at 12-14, +2 at 15-17, and +3 at 18+.
This last makes ability scores much less important in the game, and (coupled with removing the various stat-boosters) reduces the difference between a character with just-above-average stats and one with super-high stats (at least in key areas). This means it matters much less that you get the stats
just right.
The net effect of all of this is that the game can safely move to "roll 4d6 drop lowest (maybe in order)" as the default mechanism for creating new characters, which is a
major advantage for new players. As things stand, ability scores are just too important to leave to chance, but point-buy systems are really horrible for new players to grasp.
(For new players, the ideal char-gen is probably "roll ability scores, choose race & class (or, perhaps, archetype), customise equipment, play". Experienced players will probably want much more control - there's an argument that the game should include several archetypes for new players, and then an in-depth point-buy system for the experienced players to get exactly the character they want.)