This is a heck of a question! Firstly, I'd say go your own separate way. Reason being, not everyone in this band is of the same mindset. You want perfection, they're happy with a lot less. Maybe they just can't remember everything, or aren't that bothered, or aren't that good players. Maybe you find it easier to learn and remember, but aren't adaptable enough to play on after a mistake. However, I'm not saying you're right and they're not.
After rehearsing and playing with literally hundreds of bands over the years, it seems to me that some players want everything to be spot on each time, and will strive for that in rehearsals. Then get to the gig, and someone has forgotten a chord sequence, the order of a song, a special stop/start, etc. At that point, it falls apart, mainly because, as you intimate in your great question, a spanner has been thrown in the works. That apart, just how boring is it to have to play each song exactly the same each gig? I can do it, but that isn't music - it's become a trick one can trot out. More later...
Yes, mistakes do happen, for all sorts of reasons. BUT, if a song is known well enough, each and every member playing should have the propensity to recover. I have a recording of one band I worked in, and listening back, realised the singer came in late for a chorus. All the others merely played an extra bit - as one - and picked up again. No-one was aware of it at the time, it happened spontaneously. Had it been practised? Had it heck! It's seasoned players listening and responding, just getting on with the job.
So, should you 'practise' mistakes and recovery? Certainly not. And above all, don't stop when there is a mistake. Keep playing, or at least keep listening - to those who didn't make the mistake, and pick it up again as soon as possible.
More! To me, music playing is more organic than organised. Unless you're all working from an orch., where it must be followed to a T, it ought to be possible to put in another chorus, solo, stop/start etc. Even playing with guys who haven't been met before (so no rehearsals) it is realistic. It involves listening - to yourself, others and the whole caboodle. So many don't seem to be able to do that. In their own little bubble is no place to be playing in a band. It's teamwork, and getting someone out of trouble should be second nature. Singer goes into chorus at the wrong time? Go with it - keep looking and listening, but go with the flow. No point in stopping. That's making the point that yes, we did cock up. Above all, play with folk who are of like mind - one way or the other.
And if all else fails - blame the bass player...often me...