Oh, I would definitely point it out to her. Long-hand, like you did here.

I'm kind and patient with everyone except my mother, who has been nagging and hypercritical my whole life. So yes, I say these things.

"Mom, I gave you five options over the next three weeks and nothing works for you. I don't want to discuss it anymore."

“I keep giving possible times that you keep turning down. That’s fine, but if you don’t suggest alternate times, we won’t get anywhere. If this is a priority, you will suggest alternate times. Let me know what you choose.”

Op - this isn't much different than a friend

A friend who says they'll call. Or says they want to get together "soon", and then never does their part.

Take the emotion out of it
And btw, never accept emotional manipulation

I had brunch with my grandparents every Sunday morning for years. Can they have a standing date?

You should ask your kids for a couple of days that the kids are available to see Grandma. The response is "Larla and Larlo are available Saturday morning and Tuesday evening if you want to come by to see them."

If she pulls the Ethel and Laurel problem, say that you're sorry she's not available and you'll make sure to let her know the next time they are available. I would try to get the kids to commit to leaving on or two opportunities every couple of weeks, in case your mother is available. When your mother inevitably declines, then tell the kids they are free to do what they want on that day.

I’d just use the exact excuse language she used in the past with her until she gets the message.

Here is your perfect comeback.
“Sorry you can’t make it those times. I’ll let you know the next [Sat] they are free.”

That is it. No need to try to get her to see any irony or to gloat or to listen to her whining or complaining or any of that. Try to set up a time that works and move on.

I don’t even know what to say to my mom. Backstory: My mom is the type who will call and ask to see the grandkids. I will suggest a time, say Thursday or Saturday. My mom will have all sorts of plans like “I can’t do Saturday, I have lunch plans with my friend Ethel” or “Thursday won’t work, I’m taking my friend Laurel to the dentist.” Fine. But she won’t suggest an alternative, she just plays this pity party game about how she misses the grandkids, to which the cycle repeats itself.

Now the kids are teens and the tables have turned, and they now have their own “plans” with their own “Ethel and Laurel”, and my mom just can not deal. She sees this as a personal affront, whines and complains, yet won’t make any effort to see them when it works for them. Also, and most importantly, she doesn’t see the irony.

I need a comeback, something to say. Help!