I'm glad that Frankie's Girl chimed here as I think she has the most experience with hoarding parents. She also chronicled the clean out and sale of her late father's home on this website including photos. Hoarding is a horrible affliction.
I want to offer my own Mom as another data point. She is also a hoarder, also lived in her home for >40 years and let it basically fall down around her. For example, the bathroom floor rotted out from a leaking toilet so that only the toilet pipe was holding up the toilet, she let the furnace die by never changing the intake filters then threw out the repair guy (in Colorado!), the fence was rotted and falling down, the yard completely overgrown, broken appliances, she never painted anything in 40 years, the vinyl tile so worn and scratched away that you could only see it had a pattern if you looked under the cabinet baseboards, etc. She also had pet birds that she let fly around the house . . . yeah, your imagination about the bird poop situation is correct. My brother and I tried to help her, and she would be initially receptive and then get really nasty. I stopped trying to help because I have boundaries about being treated badly. My brother helped a couple of years longer and he got the rotted out floor replaced so she wouldn't fall into the crawlspace, then he also stopped.
Me refusing to visit her home (I'd stay in a motel 6, which offended her) and him telling her he was done trying to help her fix things and that he also wasn't going to visit her anymore seemed to be rock bottom for her. It took her an entire year, but she managed to cull her hoard enough that she could move. After she was out with what she wanted to take with her she told two "down on their luck" neighbors they could take whatever was left behind and sell it as long as they took it all. They kept their word and took it all out. She acts like she was doing them a favor by letting them have her precious hoard leftovers, but I think they were more happy to get her out of the neighborhood because she was dragging down their home values. She then sold it "as is" to a flipper.
She bought a new modular home near my brother. Yes, she still hoards, but it's not as bad as it was before. For one thing, she doesn't have the birds anymore, just two dogs and a growing collection of cats. I'm hoping the cats at least keep the vermin out (she had a vermin issue in her previous house.) She also lives in a tiny town now with less opportunity for finding stuff to hoard. Family checks on her daily.
EconDiva, at one time I entertained buying her a home, but now I'm glad I didn't do it. She is trashing her own new house. She tried to pressure my brother and his wife into picking out everything for it, saying things like "it's all going to be yours some day", but they refused to humor her. So, she picked the color of the carpet she is destroying. My nephew moved in with her for awhile and that seemed to help (he did stuff like vacuum and mow the lawn.) She's also run out of money which makes spending on her hoarding affliction more of a challenge. But, it is still just a gradual slide into this home ending up just like the last one. As Frankie's Girl said, it's a disease.
That all being said, I do think that overall the move was very positive for her. Like your Mom, she would have refused to go into an apartment, which she views as throwing money away. Ironic, isn't it?
If you can get your mom to move, then that will be a big step in the right direction. If you buy a home for her, though, expect her to trash it and don't get upset when she does. My advice would be to buy the absolute smallest home you can find that is in good condition. If you can get something in cash on hand, do that rather than a mortgage. Do put only your name on the title. Hiring a maid service to clean biweekly would be a big help, I'm sure, if you can afford and she will allow it (my own mom never will allow maids into her home.) Good luck!