Not telling strength or dexterity affect attack rolls has occured to me before, but ... I don't know - what _is_ affecting attack roll then anymore? Is it just luck? Do we get a weapon skill? Even with that - it seems odd.
The alternative, use different ability score modifiers, may actually be more compelling. But even then, people may be hunting for the glorious 18 or whatever, since it's still the only way to get a bonus to one of your most important attacks.
I would like to see a system where you could play a "smart" fighter and be effective, all be it in a very different way then if you are a "strong" fighter. Stuff like using ability scores as feat prerequisites or something like that. But it's all very vague and seems difficult to pull off. And even if you can play a smart instead of a strong fighter, if you're a fighter, you should need/want some strength IMO. Just not necesarily a 16+, a 13 could be satisfactory, the rest compensated by your intellect.
Maybe there should be a cap on ability-based modifiers, and instead of having one modifier rule it all, you can add two or three ability modifiers (maybe even flexibly so, some kind of build decision for characters.
Maybe one score is determined by class or weapon choice (Strength for most melee weapons except light weapons, Dexterity for most ranged weapons except heavy thrown weapons), one by class choices (Fighters may have Constitution or Intelligence, Barbarians Constitution or Charisma, Rangers Constitution or Wisdom, Rogues Intelligence or Charisma, Bards Charisma or Intelligence etc.) and maybe a third by your choice. Only prerequisite that they must all be 3 different ones. Maximum modifier you can can get might be +5, and most people would usually get that +5 in their class-typical weapon choices. But you still have the option of doing it all via one score or doing it with multiple ones.
If you want to play a party leader type of Fighter, you pick Strength, Intelligence and Charisma as your most important scores, and get a Strength of 15, a Dex of 12, a Con of 13, a Int of 12, a Wisdom of 10, and a Charisma of 14. These aren't really uber stats, but they cover exactly what your character is about. You can use charisma in social situations to act as party leader, use intelligence for planning, and strength for beating sh*t up.