As far as I know the "raining diamonds" thing is when the temperature and pressure is so high that carbon forms them.
By defintion, GETTING to the diamonds is difficult.
Since it's pretty trivial to just form lab diamonds, or find a convenient planet with accessable diamond deposits, I'd not think anyone would bother.
Diamonds aren't worth very much, objectively.
Anything that's not a gas is way, way down in the core of a gas giant. Unless there's some special unobtainium that is discovered to form there, any form of extraction is highly unlikely to be worth it.
GAS mining on the other hand, yeah, that makes sense. Don't think rocks, think methane, think helium, think ammonia. Ammonium sulfide.
You could have a station around the 1 bar pressure zone that sends gathering drones that are able to manage the high pressures down to where the heavier gasses are. It MIGHT be possible to do it with some kind of snorkel setup.