It's fairly obvious why school hours, term time only jobs aren't going to be well paid, interesting, or plentiful. Apart from actual schools, every organisation operates for longer hours than this, so if it's possible to only do those hours, it basically means that it doesn't matter if you're there or not. That means a job that is low- or unskilled, low-stakes and repetitive. Pair low skill requirements with the fact there are far more people interested than roles available, and you also get low pay.
At my company, people have all kinds of working patterns. My boss works four days, even though her kids are grown. Many people work 3 days. Our Head of HR works school hours. My first boss, one of the directors, works very flexibly; he might take several hours off in the afternoon to spend with his (adult) children, then work til 1am. (I think he's bonkers, but that's the way he likes it.) It's entirely possible to get challenging, interesting, part time work, but it requires two things: making yourself a valuable asset, and seeing things from the employer's point of view. If you can't make yourself an asset through education or skills, you can do it by proving your worth in a job through work ethic, personal qualities, ability to learn. That is why it is often easier to do it from the inside of a company.
Fwiw, though, all roles in my field are advertised full time. I've interviewed for several and have told them I would want a 4-day working pattern, and all of them have been fine with that. If I joined them and proved my worth in 4 days, I would probably have leverage to negotiate a drop to 3.
You have choices, Blair. You have chosen to prioritise family proximity and school hours and term time only over anything else, thus limiting your choices almost to nothing. This is the natural result of your choices and not some kind of conspiracy or systematic bias.