Look for care jobs?
Yes care jobs normally offer bank work so you can choose your shifts. Join a few agencies. Also cleaning jobs are you good for a couple of days a week. Thing about part time work is that employers have to ensure that what they spend on onboarding someone is going to be worth their time, so I understand why some are asking for flexibility.
I've had the same thing! Every rejection has been you are great but we need more flexibility. Turns out it's easier for my husband to work overtime and I stay at home. I still keep an eye out for easy part time jobs though and I'll keep trying until I get one x