Someone should adopt him and give him back to her. It sounds like the Rescue are being scummy about taking the dog back and trying to make money off of people by stealing dogs back.
Biggest change we can make to the whole model is making it non-profit. Hospitals should be patient first in all instances. That is probably the biggest hurdle to clear and then a bunch of the other issues you mentioned will actually be easier to solve. Right now there is too much private equity involved with insurance, hospitals and medical care in general. The costs associated with medical care are outrageous and also need to be addressed.
This article is literally 7 sentences long. 8 if you include the title. You seriously can't be bothered to click the link and read it? For fuck sakes dude come on. The video in the article is literally 1 minute and 10 seconds. You can't take 70 seconds to be more informed. You have to be a jackass?