I often hear sound guys talk of too much stage volume from the band members especially drummers, but it's not as simple as just play quieter like it would seem to be. It can fundamentally affect the entire feel of the composition for the drummer, the entire band and even the audience since the group now has less energy, especially in the smaller intimate venues where this type of problem usually occurs. Basically you are asking the drummer to tone it down, and not put nearly as much energy and dynamics into the performance as usual. Usually there's a reason these drummers play like this. I'm one BTW...Usually drummers that are really loud are used to playing with little to no PA reinforcement (perhaps just kick if even that) and having to provide the beat with acoustic drums only while competing against amplified instruments. If you think about it there is probably no other instrument that you would literally ask the person to alter their usual playing style and performance in such a big way. I guess you could go to an all electric kit with triggers but that carries a host of other trade offs.
In the smaller clubs or bars with live musicians the "PA" often isn't really needed a whole lot for overall volume, but mostly for vocals, samples and other instruments which need a boost in volume. In large clubs and especially arenas there is often enough distance and dispersion to work around loud players and the PA does a whole lot more work in these situations.
Anyway that's a subject I'm very familiar with.