I've played in a covers band which I quite enjoyed, but there were a few songs I had to play through gritted teeth, as they are not to my taste at all (e.g. 'How Long' by Ace and 'Never Seen the Rain' by CCR)
For my current band, I made a decision to only play stuff I really enjoy. It took quite a bit longer to find compared to your average pub rock band, but well worth it. Of course, there are still some songs I prefer to others, but nothing anywhere near cringeworthy (all originals now)