Of the pagans I know and have worked with in education, community and multi-faith activities, I don't find them any more dissatisfied with the education system compared to any other group or have any more issues of the oversimplified version of faiths found in the curriculum, including Christian parents - some of them are very dissatisfied. Some do home educate, but none I know for religious reasons - like most, it tends to be about local schools on other issues like behaviour or additional needs.
I could see interest in pagan online or forest courses for kids. Those have exploded in recent years, I know some forest nurseries that also run holiday clubs for school-age children, but a whole school seems like biting off a lot. Many new school end up inadequate, and that's more likely when it's more fringe whether by faith or narrow education like the issues many UTCs are having. There are also home education camps that include courses for the week or two that they run.
What you're describing sounds more like a home education co-operative than a regulated school, especially the whole 'only teachers we already know/in our group' thing. Those are legally fine when parents aren't leaving their children to be educated. Those are not daily, those are usually weekly if not less frequent. I don't think you'd get many in that with the requirements given or for residential.
A lot of Pagans gave answers like Heathen, Wiccan, Druid, or a host of other names for specific branches within Paganism.
Not all of people who identify with those would also use the term pagan or view those as 'branches within Paganism'. That's part of the issue.
There are people who have a polytheistic, animistic, or pantheist faith system who hate the term pagan because, as others said, it's rooted in others identifying others with non-Abrahamic beliefs. Modern use by some doesn't change that particularly for long standing groups who view themselves as different to neo-pagans. Plenty of polytheistic, animistic and/or pantheistic groups pray or invoke their divines in other ways - trying to claim prayer is only monotheist is weird, as there are monotheist groups who don't pray (some lean more towards silent contemplation and listening) and we have what we might now call pagan prayers from polytheistic writers in some of the oldest remaining writings we have.
I should also add that we are an all-female group, if that assuages any potential concerns.
No, it doesn't. Women can be abusers too. Celibate people can be abusers. Basic safeguarding - no group affiliation or personal traits assuages potential concerns. There is a reason why in recent years we've been getting more historic reports of abuse from closed community schools - hiring only your own is a safeguarding nightmare. It's part of why there is growing push against faith schools from many people across all faiths and none and support for governing boards and staff to be of a wide range of backgrounds. For safeguarding reasons and just to get the best people when recruitment is a major struggle, the widest net improves practice and wide range of backgrounds reduces the risks of blind spots and brings about more ideas.