Perhaps, just as an academic exercise, we should maybe arbitrarily assign a bit different methods of using the stats for determining to hit and damage bonuses. One of my major gripes in the melee system has since the early ages been the reliance on Str. I get that strong people can hit harder, but especially in armed fighting, harder does not mean more accurate (just look in system for the power attack feats). I know finesse weapons partially address this issue, but not completely.
It helps if you think about a fight not just being an exercise in chopping wood, but attempting to kill an actively-resisting opponent. Particularly when you bear in mind that in D&D, harder
does mean more accurate when it comes to the "to hit" roll if your opponent has armour/tough skin and muscle etc.
The control you have over a weapon is determined by the forces that you can exert on it. A stronger person can exert greater force. This means the weapon moves faster, changes direction more suddenly, and needs less of a swing to generate a damaging blow.
Skill is probably a bigger factor, and reflexes can certainly help - particularly when reacting to and avoiding your opponent's attacks. But even before we get into situations where you are directly exerting yourself against your opponent, or D&D's AC system where beating through armour can be part of the "to hit" roll, Strength is the secondary governing factor.
Even with smaller, lighter weapons, athleticism has a large influence on your ability to make a strike, since you will often be required to move your body quickly to move through your opponent's guard to get in reach, and then quickly move out again. (Or trying to stay within her reach as she tries to make distance between you.)
There is a point at which you have to be strong enough to effectively wield a weapon, but unless you're talking ridiculously huge and over-the-top fantasy weapons, that strength is well within the capability of adventurers. You can definitely still use a historical longsword/greatsword with a Str of 8: it will just move slower, you'll not be able to control your swings very well, and its more likely to bounce off even weak armour. There is no "threshold Str" however, at which you suddenly become more capable with it and above which you don't become better. As you become more athletic, and able to generate more power, your capability to hit your opponent continues to increase.
There is a place for Finesse in the system: its a fantasy game, and many people will generate characters based on media tropes rather than gritty realism. If you're going to start adding more complicated rules however, you need to ask yourself "What do I wish to achieve with these?" Balance? Realism? Personal preference? etc.