You don't mention what the three reasons are.
So it's hard to say if they are legit or not - from a parent's perspective.
Here's my thoughts.
1 - if it's the same practice, it's possible the dentist tries to drum up business for the orthodontist. Not unheard of.
2 - I hate the idea that the dentist would suggest your daughter have braces to make her look 'beautiful'. Hopefully he didn't actually use those words.
3 - Cosmetic reasons for having braces are not that uncommon. My son is having braces for that exact reason. The thing is, often it will correct the bite AND look better. So while my son does not NEED braces - as in, he could eat/talk/get by without braces and his bite would be OK without them (he has some overcrowding on top), having braces will greatly improve his smile and will make his teeth be spaced more evenly.
So your dentist may have a point is what I'm trying to say.
Our dentist had the opposite approach to your dentist. I really like my dentist for this reason. He never suggests anything - including braces - if they are absolutely not necessary. So he did not suggest braces to us. He did recommended a good orthodontist for us for an opinion though.
I would get two opinions if I were you. Usually consults are free or not that much money - and see what they say. If it is truly cosmetic and you can put this off - then you know.
My son was a later teen when he got his. By then, we saw what his teeth were really going to look like. Up until then, we were going to leave it, but his teeth became overcrowded to the point where he was no longer smiling. I knew by the time he had a school photo done and he would not smile - I thought, ok it's time for braces.
So - you can leave it for a while and get the opinions - or do it now, and wait ... and see how her teeth grow. I would not get braces at 10 (unless there's a reason for it). Personally, it sounds like you're being pressured, and I never think that's a good reason to get work done.
Good luck :)