I have a dd just going into year 11 too, also an only child. I agree with the previous poster that the main problem is that your dd is being too narrow-minded about which kids would make suitable friends. Clearly you have already identified this yourself, but it would be worth exploring what's driving it. It might be due to anxiety about how she will be perceived, or lack of confidence on her part, but the reality is, she won't find many friends until she is able to address it.
My dd islis very lucky, in that she has always been blessed with a facility for making friends, and she has a very wide and eclectic network. Other kids often express surprise at the breadth and diversity of her friendships. However, she would tell you that this is definitely not an accident. She makes a deliberate and active effort to talk to everyone, cool or not cool. She simply doesn't categorise people in the way that many of her peers do, and she doesn't feel that someone needs to be similar to her to be her friend. She just takes people as she finds them. If she considers them to be nice, she doesn't care if they are boys or girls, older than her or younger than her, the nerdy type, the handbag-carrying, makeup wearing type, the "roadman" type, the shy wallflower type or the life and soul of the party. She is kind and inclusive, and doesn't judge people by superficial stuff. She makes a particular effort to include new kids and kids who otherwise seem to get excluded. She doesn't wait for others to come to her, and in lockdown, she has made an active effort to stay in touch with a wide range of people.
I'm not saying any of this to boast, but rather because, making friends has never come very easily to me, and I have learned so much from watching my dd's approach. It has been a revelation, and I finally understand where I have been going wrong all these years!
DdD gets asked for advice a lot by other kids. She mentioned one girl the other day who was struggling with friendships because she felt that she could only really relate to people who shared the same niche interests and taste in music. DD was trying to get her to see that she needed to cast the net much wider, and explore friendships with people who weren't necessarily just like her. The same girl was also worried that people would not welcome any attempts to be friendly because their various friendship groups were already well established. Again, dd was encouraging her to just give it a go.
DD also does a lot of extracurricular stuff (mainly dance/drama and some voluntary work), and this has certainly expanded her network outside of school while also giving her something interesting to talk about. However, the same thing applies - just going to the groups isn't enough, you have to take the initiative to talk to people when you're there. FWIW, I don't think dd cares a jot whether it's cool to have hobbies or not, because she loves doing them.
Reflecting on my own experience and watching what I've learnt from dd, I think the biggest thing is somehow overcoming the fear of rejection. Try to get your dd to think about the worst thing that could happen if she got snubbed. Temporarily humiliating, yes. The end of the world, no. It's a risk you have to take, and it gets easier with practice.