I quite like the technique of adding a kind of "microtoast" right next to the element that's just been clicked/updated. So you'd click a button, and then directly above or below the button (or even on the button, depending on the notification), you add a bit of text saying the action has been completed. That disappears after a short delay, just like a toast. It's still got some of the accessibility issues that always come with popping up random elements in the UI, but at least it is directly next to where the user is interacting, so they can easily see that what they've done was successful, or failed, or whatever.

This works well for the last category, because it provides feedback but it doesn't need to be dismissed. But it also typically needs to be implemented afresh in each place it's used, which means more fiddly developer work.

All that says, I've lost this battle plenty of times and a lot of the stuff I've worked on ends up getting toasts in the end because they're just so much easier to implement than anything else.