I would think that you would have a choice. My vet's office has three veterinarians (the main guy, and two female (one older, one younger) vets under him). I would use the main guy for my dogs because he was familiar with them and I liked his treatment of them. I got one of the women vets (the older lady) once for a serious treatment check up on my dog (we had a flea infestation that summer and he was having a bad skin reaction to them) and I was appalled by how she handled him and how she talked to me. I will never use her again, and when I call I specify which vet I want to use. If that vet is not available that day, give me a day when he is.
My vets and their office attendants know my animals. When I am out and about running errands, I will take an animal with me sometimes. I could run up to the vets office unannounced for a medication refill, bring my dog or cat in with me, and they will know by face and name, who I am and who I have with me. It is a very busy office outside the city, and I am there once every two months.
Don't just make a vague appointment and hope who ever is available shows up. Make an appointment with a specific vet. You are the vet's client, not just the office's client. It's like any other business. They have to take care of their clientele, but if their clientele is bouncing between multiple vets (even in the same office), they can't build a relationship with you.
Think of it not as a failure but as a success in how not to do it.
Don't look in a horses mouth for a gift.