I'm not a parent, but as a kid I wasn't eligible for braces till highschool - later than all my friends. And by that time I wanted them because I was super self-conscious about my teeth.
Once her friends' braces come off and she is the only one with crooked teeth she may be feeling very differently - but if you get them now she can still moan and groan about how evil you are, and also complain with all her friends about how much they hate having braces. And then they can get matching bands for school.
Yes, they will likely hurt. Right after they're put on and a couple days each time their tightened, it's not pleasant. But it's by no means "unbearable".
I kind of think of it in a similar vein to my mom forcing me to take piano lessons for a couple years (along with various other activities). Some I liked, some I didn't. Overall I'm glad she pushed me to try things out because now I have some interests I wouldn't otherwise, but at the time I often threw fits because I didn't want to do it.
Momentary pain; lifelong straight teeth. Another analogy - if my parents could have put me on a 1-yr treatment for obesity that would have me fit and healthy for the rest of my life, I probably would have whined and screamed then but I'd be sooo glad of it now.