You're doing yourself no favours trying to paint your sister as materialistic. She deserves your gratitude as there is no amount of money that can really cover the ongoing obligations and duty she's been shouldering to ensure your mum is cared for.
As much as your mum's wishes are important, it sounds like she is utterly intransigent on the subject of having help because she knows your sister is there and this is to her own detriment because your sister can't be everywhere at once. The stress of trying to keep her happy and safe may well be unbearable and the last thing they need is you turning up encouraging her to think your sister can manage when she can't.
Remember, you swan home without changing an incontinence pad, dressing an ulcer, ordering a mobility aid, fixing her glasses, sorting out the prescription mix up, finding out why she didn't take her last tablets, arranging a check up with the district nurse, doing a wash of all bedsheets, hiring a carpet cleaner, taking her to get her hair done and accompanying her into the garden at the very moment the children get home from school and the GP finally calls. Your sister probably does an increasing amount of this on repeat for months. If she also works, she's sunk. You need to cut your sister some slack as it's hard to imagine what caring for an elderly relative can be like until you're doing it. I have done it and baby triplets would have been easier.
Really, the money is the least important thing here. Getting your mum the help she needs and making her as happy as she safely can be without jeopardizing the wellbeing of your sister's family is the priory.
It was naive for anyone to think a blanket reassurance could be given regarding residential care but it doesn't mean your sister is not in good faith. I think she's shown that she is, if your mum chose to move closer to her. Are you sure you're not just feeling out of the loop?