As a general rule (and I'm sure there are exceptions) vets are not animal behaviorists. I would not take what a vet said about my dog as truth.
A lot of dogs do not want a stranger's hand going over their head. This is a mildly threatening move, and it's perfectly normal for a dog to avoid the contact when it is presented in this manner. A person who works with dogs as often as a vet does ought to know that it is not a great way to pet a dog who may already be a little stressed out from being in an unfamiliar situation.
It does sound like you need to work on increasing your pup's tolerance for being handled. Did you bring treats in to the vet with you? A treat can go a long way towards making the exam a more pleasurable experience. I'm sure a lot of other people can chime in here with more advice on how to make a vet visit easier on your dog.
Be careful how much stock you put into the whole alpha concept. I know a great many people believe in it, and that's fine if it works for them. I didn't find it helpful to think of my dogs in this way though, constantly feeling like I was being challenged for my authority. It made me much more punishment-oriented in my training techniques, and I think this goes a long was toward weakening our bonds with our dogs. Once I set aside the idea that my dogs wanted to be in charge, and viewed their behaviors through a different lens (they were just being dogs who needed to have the right motivation to behave the way I wanted them to) their training became much less stressful for me and for them.
Classes are great, keep going to them. If you don't feel like they are working, don't be afraid to find a new trainer who uses different methods. Every dog responds differently to a training method, so what worked for my dogs might not work for yours. Don't ever use a training technique if it makes you uncomfortable or upset.
Do a little bit of research on a concept called Nothing In Life Is Free. This is more of a way of living with your dog to help her learn impulse control and how to earn rights and privileges. This is how my husband and I raised our dogs and I feel like it helped us avoid many of the behavioral issues that so many other people face with their puppies.
Niko: American Showline GSD 5 years old
Rosa: American Muppet Dog (GSD/Border Collie mix) 5 years old