Cables will always feel a bit different on drag and incline curls, but if your elbows are staying back and you’re keeping tension the whole way, the biceps are still getting the job done. Same with the French press.
It’s actually good what you’re doing with these controlled movements. Training while healing is a pain in the **** but it’s the best way to maintain progress until your hand’s back 100%.
The strap irritation’s annoying but temporary better that than setting recovery back weeks. Keep milking the machines for now, and build what you can and the barbell stuff will be there when the hand’s ready.