First, let me start by saying that I am a board certified orthodontic specialist - just letting you know where I'm coming from.
Second, I'm sure you didn't mean it to sound derogatory, but saying "just general dentistry" will likely be considered a slap in the face by many general dentists. Everyone who practices any aspect of dentistry was first a general dentist when they got their dental degree. Then, some of us went on to spend extra time, effort and money to focus on an area that we were more interested in (or maybe to avoid some aspects of dental practice that we didn't like very much?).
That being said, there are MANY general dentists who do Invisalign. For most of them, it is their main (or only) way that they offer orthodontic services in house without referring to a specialist. I'm not sure what the actual numbers are, but I think it's safe to say that at least half of the Invisalign treatments being done in the US today are being done by GPs. Many of these general dentists have done more Invisalign cases than many orthodontists (there are many orthodontists who will not do Invisalign at all!).
I agree with the prior answer that having someone who is experienced in Invisalign is VERY important. Seeing examples done by the doctor is also a good idea. Since most orthodontic specialists work closely with the patient's dentist, any other issues that may need to be addressed can usually be done by the general dentist if the orthodontist is good at communicating with that dentist and if they have together collaborated on the overall treatment plan. Communication between these two providers is key to good results!!
P.S. I have gotten into the habit of showing my Invisalign patients their ClinCheck, which shows their final results. That way, they get to see their predicted final outcome before we start! Remember that having to do additional aligners is very common (at least in my practice). Ask your doctor if that would be an option for you.