We’ve had a number of members who’ve been given that very same advice - tonsils first to see if that helps (which often it does not when the symptoms are from ES). Being able to feel your styloid during a tonsillectomy & spontaneously removing it DOES NOT sound like an optimal situation. We’ve had members who’ve had intraoral styloidectomies where tonsils were removed first to get them out of the way so the styloidectomy could be done more easily. The recovery from a tonsillectomy as an adult is pretty painful & no fun. When you add the styloidectomy to the mix, it’s a pretty miserable month long recovery.
If I were you, I’d get a second opinion from one of the doctors on our list i.e. someone like Drs. Cognetti or Samji who do video consults since both have done many ES surgeries. There is a charge, but most health insurance companies will pay at least part of the fee for a video consult. You should check w/ yours before making an appointment.
You can also post a couple of your images on here (3D preferred as some of us aren’t great at reading CT slices), & we can give our non-medical opinions. If you need help making your CT into 3D images, there is a post on our forum w/ instructions (Making Your Own 3D Images from CT Scan- 3D Slicer Tutorial) & others on the internet.