I can't comment on the stays, except is there any way you can get through to Mia that if she stays, then she can be active? No stay, no running. Like Premack?
But I can tell you that YES you absolutely can teach a dog with no toy drive to play. It takes a long time and a lot of patience, though. I've been going through that exact thing with Sienna. When I first got her she had no interest in toys at all. Well, except for one toy, which was a screaming stuffed monkey that had elastic arms, and you kind of sling-shot him across the room/yard. LOL. When I slingshot the monkey it fairly often caused Sienna to do crazy zoomies. She didn't interact with the toy at all, but she was clearly having a blast. But if I threw a ball or anything, she just looked at me like I was an idiot.
I've been meaning to write a blog post about all of the things I've tried when teaching her to fetch (and enjoy it). I wrote one about tug and things I learned at a Denise Fenzi seminar recently, which might have some insight for you:
http://thediydog.com/the-zen-of-tug/
To long story short it, I've spent literally 3 years working on a retrieve with Sienna. I've used every retrieve method, game, etc. that I could find. She now has a really high value for her dumbbell, but toys are still pretty meh. She gets excited when I bring one on a walk, but when I throw it she kind of runs in a circle and would rather play with ME than the toy. Finally, a few weeks ago, she chased a thrown toy for FUN. Like a *normal dog*! I was ecstatic. What finally made it click for her and for me, aside from the hours spent building reinforcement history for putting her mouth on things, was realizing what drive she's strongest in. What part of the prey sequence is strongest for her.
This blog post helped me see that:
http://exercisefinished.blogspot.com/2012/06/denise-fenzi-seminar-part-ii-tugging.html It's also about a Denise Fenzi seminar. Seeing a theme here?
Sienna loves to chase. Once I realized that, I tried throwing a Kong for her. The weird bouncing got her interested, and I found that throwing it really, really far was actually more interesting for her - she's more likely to chase it if she can RUN a long distance out to the toy, vs. it's just near me. So figure out what Summer likes to do best and then think about how you can incorporate that into play with her toys.
I'm still working on the very beginning of tug, using a
fox tail on a rope from Clean Run. If I drag it behind me while running, Sienna will try to pounce on it. (Which still amazes me.) And that's a starting point.
Another thing I learned was don't use food. I tried to shape Sienna to tug, and it was kind of working, but she was clearly only doing it to get a treat. Play for its own sake has to be self-reinforcing. One of the things Denise said was that you can actually kill play drive by reinforcing it with food. I'm not sure why that works, but possibly for the same reason that the Look At That game works - a behavior that was reinforcing becomes just another way to earn a click.
That's for tug, though. For retrieving/chasing, I did pretty much what you mentioned, reinforcing for any interaction with the toy. It does work, but it takes a long time.
Oh, I also used the "give me a break" game from Control Unleashed for this. I'd click for interaction with the dumbbell for about a minute, then told her "OK, take a break!" and I'd go read or something. Pretty quickly, I had a dog harassing me with a dumbbell in her mouth. (Quit laughing, retriever owners. Yeah I had to train my dog to be pushy with a toy. LOL.) GMAB works really well for teaching play, especially to a soft dog that's sensitive to pressure.