Why or why not?
When I was a kid, I always scoffed at characters with amazing abilities/backgrounds who wanted nothing more than to be 'normal' (especially girls...I can probably count on one hand the number of female protagonists I know who actually APPRECIATE and ENJOY being special and powerful). I just thought, 'Really? You'd rather be doing homework than saving the world??" It didn't make any sense to me. Why would someone with the opportunity to go on exciting adventures every day crave the boring sameness of everyday life...?
Of course, once I grew up and learned to appreciate stability, I started to understand. ^^; 'Normal' isn't just about being like everyone else, despite what most teen heroes would have you believe. It's about having some predictability and control over your life, being able to take at least a few essential things (like being alive tomorrow...) for granted.
It's also about being allowed to not be bothered. 9u9 When you're weak and average and 'normal', the assumption is that there isn't much you can do to help in any given situation. At least, not any more than any other person, so no one can blame you for minding your own business. It's easy to convince yourself that most issues outside of your own life just aren't your problem.
Not so when you have special abilities. ^^; Now, even if no one else knows, YOU know that you are more able to help than most people on the planet. You are constantly aware of all the things you could be doing, and you have to work harder to justify each instance of inaction. Even before you make the leap to 'well-known superhero vulnerable to criticism from the public', that's a lot of pressure. I wouldn't blame anyone for wanting to be rid of that burden alone.
...And yet, I still tend to give the side-eye to characters who 'just want to be normal'-- because most writers don't take it that far. =/ They focus on the cute, aesthetic, surface-level aspects of 'normality': running to school with toast in your mouth, or settling down and raising a family with a cute side character (* cough* FEMALE PROTAGS *cough *). We're expected to sympathize with the character's desire simply because of how pretty it would look, and not any of the other deeper, equally valid reasons.
I guess maybe because we don't want to address the messier, more human aspects of being a hero? ^^; We don't want to admit that stepping up and being a 'good person' is a daily struggle and maybe even a tough choice, occasionally. 'Good people' are supposed to be automatically, unflinchingly good 24/7; they shouldn't ever want to quit or feel like focusing on themselves for a change. They should be willing to give all of themselves all the time-- No excuses.
I've actually been working on a script that addresses that issue...it started out as a traditional happy-go-lucky superhero story, but with each draft it got darker and more emotional, and now it's the story of a hero who is gradually tortured and guilted into that mindset, of being willing to give all of himself all the time. He starts as a flawed but kind person, and ends as a miserable, suicidal tool...but I digress...
...Or not. =T I mean, if more heroes were allowed to admit or at least address those concepts, the idea of them wanting to be 'normal' would be much more genuine and easy to believe. It doesn't necessarily have to be dark or angsty, just...real.
Anyway, to actually answer the title question: usually no. ^^ I may have gained an appreciation for 'normality', but I never lost my appreciation for characters who are 'different' and take pride in it. My characters may want a less stressful life, or a happier life, but they rarely 'just want to be like everyone else'. And if they do want a life that would be considered 'normal', they never phrase it that way. They focus on what they want for themselves, not on what society deems an acceptable way to live.