The short answer would be no.
But that being said I can see strength being a mitigating factor in certain actions, like reducing the strain incurred for things that require brute force. ((Dragging, Smashing, climbing, swimming)) Just as dexterity would reduce the strain incremented from things like dodging.
As for almost all things tied into melee combat however, skill would be far more important than raw physical prowess for reducing the cost of actions. A trained fighter has fewer wasted movements and proper technique to rely on, that’s what makes an Akido take down look so effortless after all.
As for how “fast” someone recovers from strain, I think it should be tied into maximum health, as in a percentage of max health is how fast you remove strain per turn. Maybe 1 percent while active (Read as take any action) 10 percent while passing a turn. We don’t have a true endurance stat after all, so this way strength gets some functionality and any mutation or trait that increases health helps you shrug strain off faster.