I am afraid you are stuck., You cannot take antibiotics all the time. The ENT is not going to take out the tonsils. It is possible that your tonsils will become less of a problem as you get older. For now, I would suggest treating infections as they occur as the best option.
I had the exact same problem when I was younger (I'm 38 now). Since having my tonsils removed at 14, I have not had one incidence of strep throat. I was getting it many times a year. I was told that the bacteria can get caught underneath enlarged tonsils and it just never goes away.
That said, you're the patient and you're paying the doctor to provide a service. Interview a few doctors until you find one who will do the procedure. I'm sure you'll be glad you did. If you can't find an ENT who will help you, consider going straight to a surgeon.