My mom. 75 y.o. Widowed 12 years, lives alone. Lifelong athlete, and still fitter than most 30 y.os. Super mentally sharp.

Over the past year or so she's changed drastically. Gained 30 pounds or so. (though still fitter than 99% of 75 y.o.s) Become withdrawn, often doesn't respond to my calls or texts. My sister thinks her house has become a mess because our mom refuses to have us over, presumably because of embarrassment. At a recent family event I caught her watching movies on her phone at odd times (when she should be talking with people) - something she's opposed to. She refused to let us have a TV growing up. She's become sort of angry, always making sarcastic comments.

She's insanely prideful. She angrily retreats from any discussion of health, particularly mental health. She historically sort of views people with depression as weak, and people who aren't fit as lazy. She does not like advice from her children.

I'm torn between using the tactics of just giving her continual positive nudges in the right direction or directly confronting her and reading her the riot act, even if she stops speaking to me for six months or something.

How is this handled?

Rather than reading her the riot act, maybe start with "Mom, I am worried about you..."

Has her hearing worsened? A lot of people find social interaction harder if they can’t hear. Hearing aids can make a world of difference.

Early dementia could also be a possibility.
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Don't use phrases like "I know how you feel" "We can get through it together" Bullshit and she knows it and will retreat further. Ask her if she'd would like to talk. Ask her if she would see a therapist?

If you can convince her to get to a doctor, then do so, preferably a gerontologist. She sounds sick from your description and the dementia or heart disease (both still happen to active people) should be considered.

Been there; done that with my mother and I wish you well.

What do you mean "refuses to have you over?" At that point, I'd be using my key and just stopping by for a visit. Thankfully my parents aren't at that point yet (and they live too far away to do that anyway) but I fully expect to be having to do that with my MIL at some point. I don't care if she will think that my wife and I are checking on her. Fact is, everyone needs other people around to function "normally" and it sounds like your mother is starting to lose that.

Sometimes in life it’s best to just accept that there are things that are completely out of you control.

This may be one of them.

She moved there a couple years ago, and I think doesn't want to admit it was a mistake.

Sure, it might be. But I want to at least put an honest effort in first.