Sounds like a wacko to me

I mean, who picks up stray dogs, gets them neutered, and then puts them back out on the street for someone else to find

She's grasping at straws!
No microchip, no tattoo-----no legal way to prove that the dog is hers. You don't have to meet her at the vet, but honestly, if you do it might stop all this nonsense, though the vet may not be able to tell how old the neutering scar is. I've heard that when looking at photos to compare dogs, to look at their nose. It's as individual as fingerprints and could be a good way to identify the dog.
I know you're trying to do the right thing and find this dog's owner, but as crazy as this person sounds, I HOPE it's NOT her dog!!!
Good luck