I have run out of continues more times than I can remember, and I have stopped playing games after I have run out of continues. Sometimes (frequently, especially as a kid), I would almost immediately restart the game and try again. Sometimes I would take a break from the game and try it again later. Sometimes I would give up and stop.
Importantly, though, I don't think I ever stopped playing a game because of the continues. Instead, the definitive "end of the run/game" message being out of continues sent me gave me a chance to reflect on the game and whether or not I was enjoying it. If I walked away from the game as a kid, it was because I was not enjoying it. Nowadays as an adult with lots of obligations, I may actually stop playing a fun game because simply I don't have enough time, but even then I don't think I'd consider the limited continues the factor that made me stop; I would have been making the time-value assessment about my gameplay either way.
In this sense, I think I may actually like limited continues, because they force self reflection. Don't hold me to that if I lose in a frustrating way, though!
I am fine with achievements as an optional feature. There are plenty of games I played where I learned more about the games because of the achievement lists, or when I got a laugh out of a hidden achievement. I don't know that I've ever played a game to 100% it on achievements, though.