Part of the issue is compounded by unavoidable stuff depending on the play style. Unless you treat the outdoors as poison or exclusively use magma traps, dismembered enemies and rain are unavoidable for instance (though you can turn the sources of these off, not everyone wants or is willing to do so.) But that's neither here nor there.
Some get around the issue of martial prowess needs with marksdwarves, but to maintain huge numbers of marksdwarves, a segment of the population (be it weaponsmiths and wood burners/furnace operators if lacking magma or craftsmen,) has to be dedicated to making nothing but ammunition for everyone else to train with, never mind fight with.
This also will bring to mind access to weaponry at minimum, though that can be handled with simply having dwarves get weapons from the stockpiles and returning them when they're done (much like children do with toys, and drinkers do with goblets if the tavern's stock of them is filled,) preventing issues concerning that, and if there are none, they can still practice wrestling.
However, I give this my +1
This solves a major issue civs like dwarves and some humans have, makes use of existing systems, and wouldn't be very intrusive.
Dwarves using a gym from an active barracks can also watch sparring matches between soldiers as a mood boosters (Urist was delighted after watching a sparring match,) with less combative sorts possibly not enjoying seeing such things - someone who finds skill with weapons and fighting distasteful for instance, won't have a martial prowess-based need, and won't like seeing sparring matches (Bomrek was annoyed after watching a sparring match.)