
You are entitled to a complete puppy health check when the vet examines a puppy prior to vaccination. My vet spends ages checking pups prior to vaccination, and also checks the eyes for me. I expect a vet to check that a puppy is well enough to be vaccinated, and I would be spitting feathers in your situation. My own vet checks my adults prior to boosters, and will delay jabs if a dog is off colour.
I agree that you need to change vets, but you have an issue to deal with first. Having paid for a service, you need to sort this first. Even if it means speaking to a senior partner in the practise.
My own practise started a new procedure which I was never happy with. Full health check and first jab with a vet. Second jab with a veterinary nurse. I took a very small puppy for a second jab, and the nurse stuck the needle through the pups scruff, and shot the vaccine down my hand. And didn't bat an eyelid!
The following week I saw the head vet in the small animal department, and mentioned that I was not happy that my pup had received sufficient vaccine. He didn't seem too happy that I had complained, but tough! He told me to book an appointment with the head nurse, and my pup had her second jab, and her card amended. I understand that the head nurse took a session with her nurses, telling them that they had to report any unsatisfactory jabs!
I have never accepted a nurse appointment for puppy jabs since! If questioned by the receptionist, I just smile and say 'I always like a vet please.' My average vet bills at the practise is over £4,000 a year, although has been double that in a bad year, so I assume that they wouldn't want to lose my custom.
Remember that
you are the customer. I would not accept being fobbed off. Once you have had the second jab, ask around for recommendations in the area - and change vets!