I think the idea of using coarse grounds for the French press is mostly for filtering out the grounds with the steel mesh on the plunger. If you have a finer mesh, by all means, go for it.
I’ve also seen some of my fellow coffee lovers at work use filters wrapped around their press, or using an ultra-fine metal mesh on their air press. They’ll go for a finer grind, and it definitely helps the brew steep quicker.
Personally, if I’m using a glass press, I can usually tell the brew is done by jostling the pot and seeing if the grounds sink. In my metal one, it’s usually 5+ mins, and then it’ll jostle and test the resistance with the plunger to see if the grounds sank.
The real answer is to have them put their heads together.