My girls started competing at age 8 and I really found it only did good things for them. Having a program to work on helps them focus on new skills in a much more palatable way than just doing them over and over again, and it is fun picking music, dress, and all the other creative decisions that go into competition.
My girls are both very competitive and can tend to focus on the negative--who got what honor at school they felt they deserved, etc. I was worried how they would react to not winning every competition (because they are perfectly competent but not skating prodigies by any stretch of the imagination). But I think doing competitions and sometimes winning, sometimes coming in last has been good for giving them some level of "immunity" to both the highs and the lows. Perspective, really. And the basic skills competition groups are small, as others have said, so you don't have the pain of being last of a huge group. Often they give medals for 1st to 4th, and sometimes that's the entire group. But at our state games last year, both my kids came in last of 8 in their respective groups, and I was really impressed with their attitude about it, which was "well we're going to work hard, come back and kick butt next year!" No tears.
I do have to add to all of the above that competing was entirely my kids' idea. It's been on their radar since they were 4 and saw their friend do a competition at the Snowplow Sam level. And it's amazing to me to watch them when they get out there on that ice. Neither of them is tremendously outgoing in "real life" and neither is a super-strong skater, but they both hit the ice and really perform! Smiles, energy, and a level of enthusiasm that really makes my heart swell. If your daughter wants to do it, I bet you will have a similar heart-swelling experience!