I just wanted to give my sympathies here ... it is really difficult to care for someone like that.
My own mother has been diagnosed with terminal cancer for 8 years, and during this time both her demands and attitude are just getting worse and worse. I could imagine my own mother saying something very similar, but that is the kind of person she has always been really. Her comments, whilst sometimes quite inappropriate, usually come from a place of love ... and frustration, though. She has very different ideas of how to portray 'love' than I have managed to teach myself. Yes, it is not easy to come to terms with this. Yes, I do still struggle a lot on an emotional level.
Whilst I agree that your mother's words were entirely inappropriate... could they perhaps have actually meant "you really should just go out and just enjoy yourself sometime" ?
Does she appreciate at all the impact that caring for her has on your social life?
I've never had to be a 'full-time carer' for my mum, and thankfully she is still able to wash and dress herself, but my fathers carers have added her to their list to check up on her when they are there. I don't live with her either.
You say carers come during the day whilst you are at work. Could it be arranged that they also come at weekends to allow you to go out?
Whilst not specifically 'caring', I spent many years trying to help both my parents when they first fell ill 8yrs ago (appointments, physio's, day-trips, odd-jobs, shopping etc..) that my own life seemed to get put on hold. COVID helped to give an excuse for a break, and then they moved a few villages away so we're no longer quit as accessible as '2 minutes walk away'. But I've still found it very hard to mentally and emotionally detatch.. Now, 8 years later, I am trying desperately to rebuild what I once had.
Please don't be another Me. You really do need to sort some additional help or respite. At home perhaps, if you feel a care setting gives increased risks? (ie. COVID etc..)
I would definitely investigate the Attendance Allowance though. My parents have never been 'high earners', but somehow seem to have wrangled no end of assistance. They get 'sits', where the carer basically just sits with them. And 'social hours' where they take them shopping or out for lunch etc..
Sorry, I will end my ramblings now, but I just wanted to send some support your way!