Just a note from a vet's wife on this... When the doctor takes an animal out of the room to draw blood it's usually one of two things. Either the doc knows the pet's owner is squeamish or overly protective ... or, that doc doesn't know how to find a vein easily. It's harder with fur folks!
My husband never removed an animal from the room unless the owner didn't want to see or he felt they would over-react in some way.
OK, I'm bragging, but vetman was and is the best veterinarian I've ever seen. Alas, he's retired.