This is something I have very strong views on. I never, ever play a gig with someone else's kit. It's just a lazy sound guys excuse often used by crappy PA companies who can't / can't be bothered to re-EQ a kit between sets. There's little, if any, difference in setup time between changing the kits over & modifying what's already there. It's dead simple. You have your kit side stage ready to move into place. With two crew, even with a fairly big kit, this takes 2-3 minutes max. I've done this many times at some fairly big gigs and never had a problem.
Well, good for you, but this policy simply wouldn't work for me, and would definitely mean I'd have to turn down gigs. It's not just a matter of re-micing a kit, it's also the fact that the other kit(s) need to reside somewhere when off the stage, and smaller venues usually don't have the room for that. And "crew"? I guess you're at a higher level of gigging than many of us... I certainly haven't had crew members on any gig I've played.
Lazy sound man? Well, maybe, but I can't say I blame him. Having to soundcheck two, three or four kits instead of one would take a lot of time, and it's not like the sound guys on small gigs make a killing doing their job as it is.
As for the actual swapping of kits... Well, on a regular small gig, about half the audience disappear between bands as it is, either temporarily or permanently. Removing the microphones from a kit, moving that off the stage, getting another kit on stage, placing the microphones and then doing a quick sound check will take
at least ten minutes extra, and probably closer to twenty-five more often than not. If I was at a bar gig with a few bands I'd never heard of, and I had to wait up to half an hour between bands, I'd probably lose interest. At the very least, all the energy the previous band managed to project to the audience will be g-o-n-e.
Separate kits are great for bigger gigs, where you can store several kits partially or completely set up off stage, where the soundman and his equipment is good enough to quickly get a good sound going on a new kit, and where there might be people who can help you get kits broken down and set up, but on a small bar gig, with two, three or four bands sharing the stage on a single night, it's just not practical.
It sucks a bit for us drummers, because we're the ones who generally have to be stuck playing other people's gear in these situations, but I think we come off stronger as a result of it. In addition to this, I find myself playing different stuff simply by sitting behind another kit than I normally do, so the whole situation might lead to new and inventive playing