While I would definitely agree that there are a great number of situations in which the necessary course is to persevere, even if it means not being able to turn in a performance of which you'll be able to say anything more positive then 'I made it through and we didn't have to cancel the gig.' I also think there's a necessary out you have to be willing to take if the cost of getting through the gig might mean irreversible complications down the line or if your playing is compromised to the point that the performance will be so bad that it's not worth the negative repercussions. Scrapes, bumps, bruises, cuts, the occasionaly projectile vomiting, sure, I undersand that. Performance anxiety? I get that too, actually, having found that after years of performing I started getting panic attacks and nausea before every gig (which turned out to be related to the dynamics of the band I was in...simple solution: quit the band. Problem solved.) But it's a balancing act. If I'm going to potentially do, say irreversible nerve damage or something than no, not going to happen. If my wife or my parents go into the hospital, then I'll drop everything because that, to me, is more of a priority.
Mind you, these are all extreme situations and I would imagine that most would be in agreement that if it's a choice between a possible career (or even hobby)-ending injury and playing a short set at the local brew house we'd all opt for rescheduling where possible. Like anything else it's a matter of balancing short term vs. long term with a bit of common sense.
That said, I definitely think you've got reason to be upset with your singer and maybe a serious Conversation is in order. Your reputation will in part depend on the band to which you're linked, and if one member isn't pulling his weight for any reason, that'll reflect on the band as a whole. You don't want your reputation and professionalism to be questioned just because someone with whom you're associated is having difficulty (for whatever reason) maintaining his commitment.
J.