Hmmm, I agree with you Jas that some of what your vet is saying is quite different than what many other vets/books would say. I'm certainly not a vet, but I'm definitely a crazy cat mom and as soon as I got my cats four years ago, I bought every book out there on cat health and read them cover-to-cover! So I promise you I'm not just making up my advice!
I just don't see a reason why the kitty couldn't be tested for urinary-related issues (kidneys, etc), and for diabetes (this can cause excessive thirst and urination), which are relatively cheap tests, but as I said, I'm not a vet!
In terms of picking your kitty up and putting her in the box, I had to train one of my cats to use the box when she was young, and this is exactly what I did. She'd still be pooping on the floor today if I hadn't have done this!
Is your kitty pooping in the box? Or not using it at all?
Maybe you could try getting a second opinion online? There are quite a few sites where you can ask a vet questions for free or for a small ($10-ish) fee. Explain everything that has happened just as you did here, and explain that you would like to try behavior modifications first. Don't mention what your vet said (cuz that might sway their response), and see what they have to say?
Finally, if you think she is stressing out about something, you could try getting a Feliway diffuser: FELIWAY - Official Site
They are also available on Amazon and probably in some pet shops. Many people swear by these for reducing stress levels in cats.
I really hope things work out!