I work three days a week (UK). It's totally possible but you're right that it does constrain where you work. Smaller companies where you can talk directly to the boss is the thing. I have a family caring role so for me it's pretty non-negotiable. I just can't take most offers (which you're right, usually start full time even if in reality there's flexible working available to permanent employees).

It would hugely open up the field to women with caring duties if more companies would do this. Perhaps this is why they do not - I don't know.