I've read this - as a not uncommon situation. That a relative afar not doing daily support doesn't believe relative (who's been doing that support for years)and not fully understanding how disabled & level of needs parent has become or how it may be extremely difficult when parent expects relative to provide all their care 24/7 and refuses equipment or carers to assist.
When you get a social worker assess , they assess needs talk about realistic options , talk with client and it's an independent voice. I think that might help in this situation where relatives may have different perceptions of mums care needs. Ultimately if mum had no legal part ownership of their main carer (daughter)'a house, then her right to continue living there is limited to being 'with agreement of whose house it is', without anAST tenancy which you cannot get for living with a resident landlord.
As I said before - 7 years of support living in someone else's house is quite considerable and would be costed quite highly.
So for eg living in a shared lives (adult placement) house with meals drinks meds done and general supervision & taking out a bit, would be in region of £465- 565 a week plus three days a week if day care (so £200-300 pw) in our LA and long term and arrangement for rent (but mum paid for extension so you'd expect no rent) .
Now times that by 7 years x12 months x 565 and you can see how quickly she'd have burnt through her money as a self funder. £6,800 a year £48k for 7 years. That's before you include additional carers for personal care etc help and possibly 24/7 assistance with supervising mobility to and from toilet etc as and when her needs increased, which might have escalated at some point to nearer to residential placement or live in carer level - so between £750-£1250 per week. At live in carer level as a self funder mum could have spent £70k in a year alone on care. That your Dsis has done for free or just for £90 per week attendance allowance.
Until you have that independent needs assessment if would be difficult to say what mums needs are, I would think the hospital discharge teams have given advice. If they have said mums needs are at residential care level, given they look to alternatives to keep people home, then that's a bit of a heads up.