Also out of interest what would my options be if I were an aristocrat (patrician/equites) with more money at my disposal?
The findings of bioarchaeologist Marshall Becker demonstrate that the ancient Romans were quite capable of extracting human teeth with skill, even if scientific analysis shows that the teeth themselves were not particularly well-cared for.^1
This page from the British Dental Association, which unfortunately does not provide any sources, says that regular physicians were the ones who performed extractions.
Access to care didn't seem to differ greatly among the classes, except maybe by price. In The Natural History,^2 Pliny the Elder talks about certain unscrupulous practitioners of medicine who charged exorbitant sums to the wealthy elite in return for acting as their personal physician, but takes care to note that greed in medicine is the fault of a few individuals. Plutarch makes note of physicians in his work, saying that their lack of scruples has more to do with how they try to get patients than how much they charge.^3
My interpretation of all of this is that the patricians paid their own private doctors who treated them, which might have translated to better care; the rest of the doctors presumably offered their services to the public.
Sources: