They should also have vet records of hip problems to help with that if this is your regular vet , they should know where your dogs health is. I'm sure this assistant is an extreme animal lover , I know it bothered me to see people coming in to put their dogs down while working at the vets office when they seemed perfectly healthy (I had to realize I didn't know the dog before I opened my mouth but it IS hard). They are there to try to SAVE lives not take them. I wouldn't be too hard on her...she just cares for the animals she is seeing coming in which IMO is a good thing. I mean you can't begin to understand (unless you worked at a vets office) the people that come in there who HAD to have this dog or that dog but cannot afford the treatments or sometimes even just the shots needed for them and end up tied to a tree outside for it's entire life...it's just SO sad. She probably sees this on a daily basis. ( I know she should not treat customers that way but I try to give people the benefit of the doubt and try to see her point)
I feel bad for your girl that's for sure but like you said, your family knows her best.
We just recently had to have our beloved golden retriever mix put down who was close to 20 yrs old and the whole family went into the vets office to be with him. They knew us well and also had all the records so no questions asked at all (he had already been through many treatment options by that time) We all said our goodbyes and all stood around him talking to him. My husband carried him out of the vets office while my daughter, son and I all followed crying our eyes out. We took him home, dug a hole and buried him in his favorite blanket. It was horribly sad but peaceful at the same time. Good luck to you and to you pet as well...hope things work out. Keep us informed on how it goes. Glad you were able to talk to the vet he IS the one who knows your dogs health over the assistants.
I also think your dog deserves to have someone he really loved be with him during this time.