Well yes, you want a used MTB. Craigslist. An upright position is not at all what one needs - that's the worst thing for a bad back. The upright position puts compression loads on your spine when you hit bumps. That's bad. Leaning forward puts your back in flexion, which is what you do want. The reason you don't see bikes like you are describing is because they're a bad idea.
I have a primer on bike adjustment here:
https://www.bikeforums.net/21296948-post3.html
This is written for road bikes, but MTB fitting is the same, substitute "bars" for "hoods". As you say, you can adjust bar height with stem angle, but watch that your upper arm/torso angle stays at about 90°. Never adjust reach with saddle position.
You won't find an inexpensive used MTB with only one ring in front. No big deal - shifting the front is not hard if it's adjusted correctly.
Ask you doctor for a prescription for physical therapy for your back. Almost all back issues can be fixed enough to ride a bike with therapy. For starters, just go for walks with your dog. As you walk, consciously rotate each hip in a circular motion so as to flex your lower spine a little. For me, that hurt like the very devil for the first mile and then got better. Eventually I was walking with no pain at all. But talk to your therapist.
And no, don't ride your bike with your dog on a leach or harness. Especially with a bad back! If you want your dog to pull, get a dog cart.
https://k9carting.com/
If your dog is bred for pulling, it'll love it. You can teach it the voice commands. I think that would be a gas.