Some dentists do the surgery but they need to be qualified with the credentials and experience.
So technically you don’t have to go to an exclusive surgeon. Your dentist is referring you elsewhere because he/she likely cannot perform the surgery.
If you do find a dentist who does it, then do your research to ensure that he/she is licensed and preferably does not have any record of complaints or formal actions against him/her, which at least in the US can be found via state doctor licensed-professional databases.
Also check if he/she is board certified. You want to make sure that the dentist/doctor operating on your mouth is accountable to someone.