It certainly feels more "realistic" to expect a giant to hit hard rather than hit more often, but that's mostly because of their size rather than strength.
But alas, the fundamental rules of the game say that Strength applies to melee attack rolls... I don't think we want to start changing that, but I wouldn't mind having belts that work like you suggest. OTOH I wouldn't mind either if they worked like now. Higher hitting rate is not that different from higher hitting damage, they both equate to killing monsters faster. There are some corner cases of course, but the main effect is not that different.
Well, it worked out nicely until 3e. In AD&D you got a to-hit bonus and a damage bonus from high strength, and they weren't the same. The highest strength you could start with was 18/00 and it gave +3 to hit and +6 to damage. 25 strength - which was the max - gave +7 to hit and +14 damage.
In 3e they "streamlined" stat bonuses to be (stat-10)/2 and you applied that to everything. In addition they inflated the strength scores of monsters. The Titan went from 25 strength and +7hit/14dmg to 43str and +16hit/dmg. This is the problem.
My conclusion: we don't really need to remove strength bonus to-hit, just going back to the range we had in AD&D would alleviate the problem to a large degree. As you noted, the rapidly increasing damage bonus from high strength feels appropriate, while the to-hit bonus doesn't.