She may not be wanting to ‘steal your life’.
Like most people, she’d probably like to live nearer to you, so she can see you more easily and more often. And know she has a loved one close by as she gets older and needs more support.
However, she’s being unreasonable to expect to insert herself into your life so fully by moving to exactly the same place and expecting to come with you to your hobbies and social events.
IMHO you should separate out her expecting to do that and her rudeness. Even if she were charming and enjoyable to be with, wouldn’t you find it suffocating for her to join your social life and expect to do so much with you?
However, I agree that her rudeness would likely result in people distancing themselves from you, if you came as a pair. So would be detrimental to you.
IMHO you should let her know it’s not fair or reasonable for her to expect to join your life: she needs to build her own. But you could support her.
If she’s been abusive to you in the past, you’re right to keep a distance. If not, she’s your mum and IME we all get to a certain age where we should be there for our parents in recognition for all the sacrifices and support they’ve given to us. It’s easy to lose confidence as you get older. Your life flies by and you suddenly find yourself pretty lonely, especially in retirement. Some people who are depressed, unhappy, lonely and lacking in confidence can become grumpy and abrasive. Life is a series of stages we have to adapt to and it may help her to see herself at a new stage, with a fresh start.
Could you help her to join clubs and hobbies to make friends where she lives? Could you suggest she thinks about how she interacts with people, including via therapy, so people enjoy spending time with her? Has she looked at volunteering opportunities?
If she’s wants to be closer to you, and a fresh start, as a compromise could she move closer to you but not to the same place? Could you help her consider options of places where there are lots of opportunities to meet other people or pursue her interests - or get into something new, e.g. an art or book club, where she has a focus when alone and can also join with others. And where there are transport links to where you live.
Ultimately, she’s responsible for her own happiness and social life. She shouldn’t expect to be dependent on you.