Where do you live? I have had three different dentists give me gas, and they all do it themselves.
The only time I was completely out for a procedure under GA was my wisdom tooth extraction, because that was actually a pretty involves surgery; they did all four at once, so as not to put me under twice, removed a bone spur on my jaw, and one tooth that had invaded to roots of the tooth next to it, which took a long time. I was under for 2 & 1/2 hours. An anesthesiologist was present at the oral surgeon’s office for that, and I did have to see a maxillofacial surgeon for this-- an ordinary dentist could not do it.
But I have had one root canal, and one marathon session where four childhood fillings in different places in my mouth were replaced, and those were all done under nitrous, not GA.
I always have nitrous at the dentist, even for cleanings, because I gave an extreme gag reflex (I could never do porn). I have thrown up on dentists before. The nitrous prevents this from happening.