I would ask for a second opinion tbh - my dd had the same problem and the orthodontist was happy to wait for the baby teeth to fall out on their own - which they did and her adult teeth are absolutely fine, she won't need braces (she's 12 now)
dd1 has had two teeth removed under GA. It was worst for me than her tbh (had to hold her whilst GA went in). there was panic when she came round and saw bleeding but otherwise recovered really well. we were on the childrens ward for 6hrs max.