Stripped ceiling, patches of paint peeled off...

I stripped ceiling paper with a steamer and underneath is one coat of paint. In some places this seems sound, in some places there are tiny blisters, and in some places large areas (1 foot+) have peeled off with a rubber-like texture.

Underneath the surface seems good and smooth, but is very slightly powdery. Distemper? Surface is grey plaster as far as I can tell.

I'm presuming paint is coming off due to distemper and/or not using a mist coat?

Do I need to get it all off, or can I sand, prime and "paint in" the bare patches? What is best?

If it all has to come off (please, no no no!), what might be the best method? Steaming worked (too) well in some places, but other bits do not look willing to come off...

use Zinsser Bullseye 1.2.3 Water Base Primer sealer that`ll hold on to the bare patches........then a fine filler and light sand .......Just done same myself .....job`s a good `un

Thanks for the advice Sounds like a bit of a job - since I'm planning to get the walls skimmed, would I be better off just getting the ceiling skimmed too? If so, what prep do I need to do?