I don't know how old your mom is, but when I was approaching 50 and well into perimenopause (the gradual shift of hormones toward full menopause), I was suddenly burdened with emotional issues, low energy, and new health problems that I had no idea would be waiting for me. I had always been so strong, capable, and there for other people. What a shock, for me and for my loved ones.
For the last 15 years, I've been mostly in the place that your mom might be in. I can rise to the occasion when heroism is called for, but honestly, I'm tired, I live with constant discomfort which restricts my ability to bend, lift, or move quickly, my tolerance for lots of activity and noise is considerably lower than when I was younger.
When my daughter decided to get pregnant almost six years ago, my first worry was that she'd want me to babysit. How could I, when getting through my own day was such an effort? Fortunately, I fell so in love with my newborn grandson that I make an effort every week that is for me heroic, but rewarding for both of us.
But what if I didn't feel that connection ( which, really, didn't have to happen – we don't necessarily choose whom we'll fall in love with). Then babysitting would be a chore, which I'd struggle to find the willingness to do.
You and your children "desperately need" your mom's support and love. Perhaps she is desperate for the same from you. The fact that your kids want to see her more speaks well of the love she gives them when they're together. Could be that handling three autism spectrum children at once is simply too much to ask of her. I'd be terrified, myself, that I wouldn't be able to keep them all happily engaged and safe.
Can you give her the option of spending time with them one at a time? That might be a win for all of you.