How do I get my mother to accept the fact that she needs to have a caregiver?

So are you saying that your idea didn't work? Cause it sounds like a good idea to me telling mom that her caregiver is a 'friend' of yours.

I have found that my mother is open to ideas when she thinks she is helping the other person as much as they are helping her. Can you tell her that you would like for this friend to stay with her when you are gone for your mother's well-being and so that the friend can have some company also because she is lonely or needs company too?

I shared with my Mom that her caregiver's husband just lost his job and she was the sole breadwinner (all true, by the way). The caregiver is really a nice young woman, so Mom took to her immediately.
It was really hard for her to accept someone in her home at first. Now, it is much easier. Of course, I screen carefully and assess personalities. For example, Mom appreciates someone who is upbeat, but does not like excessively chatty or nosey people.
You have to set yourself up for success when you choose an in-home caregiver.
good luck