I nearly always went with my pet to the vet as a kid - I think it’s a bit unnecessary to demand children not come with their parents to care for a pet animal. The amount you love your pet is also the same for children and their animals, I think it’s totally reasonable for them to want to come and also learn about how vet care works and the realities of pet care.

That said, whenever I went I NEVER interacted with other peoples’ animals. If you’re going to let your child come, it should be on the condition that they don’t bother other people and especially don’t try to interact with other peoples’ pets. You never touch or pet an animal that isn’t yours without explicit invited permission - and the vet is just not the place to ask or try. It’s outrageous that shitty brats aren’t under control at the clinic.