Welcome to the forum, I'm relatively new too and have found this forum a godsend! What a scary situation for you. We were in a similar situation with our relative sending carers away. We pointed out to the Social Worker that our carer was at risk from not eating and neglecting personal care, that she didn't have insight into her condition, and therefore did not understand the full implications of refusing care. Social Services have a duty of care to your mum, which includes ensuring she is adequately nourished and that her basic personal needs are met. You could point out that by taking your mum at her word and leaving before preparing food for your mum and reminding her to eat, the carers are putting her at risk of malnutrition.
Social Services do have the power to provide needed care even where it is verbally refused. I am sure you are not asking them to instruct carers to drag your mum kicking and screaming up or down the stairs, or force her to eat - this would be counter-productive and only result in traumatising your mum. It sounds like the carers don't feel empowered to try to be firmer in trying to get your mum to accept care, so you need to get Social Services to accept that care must be attempted even when it is against your mum's verbal refusal and instruct the carers accordingly. It still might take carers some time to gain your mum's trust and actually achieve anything, but if they take a firmer approach and try to persuade her, they will eventually get there - it just won't happen if they don't even try.
We had meals on wheels being sent away. We had ourselves offered food, which was refused, but when we left it in reach, it would disappear in 3 minutes flat! Our relative is self conscious about not being able to use her bridge, so refusing food might have been about worrying about being observed eating. We insisted that meals on wheels had a key to let themselves in, ignored her shouting at them to go away, and put the food within her reach. She ate the meals most days and later told a carer she really liked them - we even found a diary entry in which she said how much she liked the meals!
She was lucky enough to have amongst the agency staff one excellent carer who gained her trust after 3 months of persistence. The carer is now her best friend, and it is truly lovely to see the affection and banter they have - both have a dry sense of humour. The carer gained this by being no-nonsense but without being bullying, there was real warmth there.
Once you've won the battle to persuade Social Services to provide care against your mother's verbal refusal (and I don't believe that refusal is the same as 'wishes', as wishes are considered and include an understanding of the implications), you have a longer-term battle to get your mum into a care home.
Document and date every incident of unsafe behaviour, falls, missed meals etc. Establish communication with the carers, and get them to report any unsafe incidents to Social Services and you - they will be all too happy to do this as it is distressing for them to observe someone who is at risk. Periodically write a letter to Social Services (keep a copy) outlining the safety concerns with a list of dated incidents as evidence. Ask how they are fulfilling their duty of care to your mother by taking steps to assure her safety. If they claim the care package is doing this, you can then show how the care package is failing.
In the mean time, start researching care homes, and take your mum 'out to lunch' at any you think she might like - she may surprise you if there is a nice one. You won't persuade her overnight, you need to start planting seeds. I think people cling on at home not only for all the obvious reasons, but also from a lack of knowledge about what a good care home can be like.
Best of luck!!!
Dagne