A few years ago, I taught a child with severe medical needs and disability- he needed 2 TAs to keep him safe in school. We had 3 TAs plus a few other members of staff trained to deal with his needs, so there was cover in case of staff absence and to ensure TAs got lunch breaks etc. It would have been impossible to manage without TAs- a teacher could not manage his medical needs and keep a class safe. These TAs, in my opinion, worked in a very high stress role, as unfortunately, this child could have died if medical emergencies were not responded to correctly- I couldn't have done it!
I've also taught students with physical disabilities which meant they needed help with toileting etc, again, this needed to be done on a 2:1 basis- and at secondary level we really needed same sex staff to do this for the dignity of the students. Recruiting TAs is difficult- recruiting male TAs is even harder! Some of these students also needed support navigating the school building- and again, this isn't really something a teacher could do, even though in some cases it's possible for the teacher to support students appropriately in lessons.
And there are also students with sensory disabilities, who again may need help accessing lessons, or navigating the school site etc.
All of these students were academically capable, by the way and suited to being in mainstream education with the right support.
Schools do have increasingly limited budgets (and don't forget the amount they get per pupil varies between LAs). Even with an ECHP, schools have to fund the first £6000 in support from existing budgets- but in some LAs, the funding they get per child is under £5000- creating a huge shortfall. And the LA funding will usually assume a TA will be paid minimum wage, or very slightly higher. For schools to offer more money to TAs, in these circumstances, they'd have to take it from a different pot- so other students would lose out in some way, because it's not like schools are rolling in spare cash.
Schools are facing big increases in bills this winter, as well as having to pay more for everything due to inflation. There isn't spare money in the budgets- this is why the education unions (and the NEU are campaigning on support staff pay as well) are saying any pay rises must be fully funded by the government.
Term time only work is only relevant to a certain subset of the population who have kids of primary school age- but secondary schools often start earlier than primary schools, and TAs may be asked to start work from e.g. 8.15 to attend staff briefings, or do handovers with parents/transport etc. So then, they may have to use breakfast clubs etc for their own children. If they are asked to support a child attending a club or extra curricular activity, they should be paid for this time, but again, if they're doing the job to work around school hours, it no longer works well for them.
Financially, it may make more sense to work full time, and either pay for holiday clubs, or split holiday childcare with a partner etc. And as more people work from home, it can be easier to have older children in the house whilst a parent works.
In the past, I think a lot of schools relied on people wanting the term time only, school hours work- but the TA role has often crept up, so it's not school hours only, and there's perhaps less need for term time only. If people can afford to work part time, there are other jobs which are less stressful!
I know a lot of secondary schools where it's very difficult to recruit TAs (and in FE as well)- often, the class teachers are asked to do more to support the students, but this doesn't work if the child has physical needs which affect them outside of class time too. And the increased workload for teachers can contribute to teachers burning out and changing jobs too.
Something in education (i.e. the funding) needs to change very fast, or I really think we'll see schools starting to crumble due to lack of staff.