It is hard but it's really not unusual.
I do have some 'Mum' friends but they all work and have children that need putting to bed in the evenings too - the chances they'd be free to provide childcare at a week's notice in the evening when it's not an emergency is remote.
My parents are elderly and infirm to the point of being housebound and live far away, some of our family and friends are abroad. My longer term friends all live 4+ hours away because we've moved for DH's job. We're really not unusual in not having family or friends nearby to step in at a week's notice.
In an emergency - e.g. if one parent was seriously ill - people would rally round but I think it's a bit much to expect other people to rally round like that and put themselves out for a planned work event where the onus should be on school management to provide sufficient notice.
TAs, including HLTAs, are paid absolutely terrible wages for the level of responsibility they have and recruitment is almost impossible as in many areas you get paid more shelf stacking at Sainsbury's. So school management really do need to go the extra mile to support staff with out of normal hours training etc if they want to keep their staff. Giving at least a months notice should be the bare minimum. TAs aren't paid enough to cover emergency childcare rates.
There is no way I'd accept this level of notice for an out of hours event for a job paying a TA wages. If I was in a 40k+ job then I'd pay for emergency childcare. The type of job and pay of course makes a difference. If OP can afford to walk away, that's her right. There are plenty of jobs out there that pay as well as TAs with lots of vacancies at the moment.
The idea of leaving a 3 year old with a random stranger 'neighbour' just to do a TA out of hours training at a week's notice is laughable. Clearly the OP is not going to inflict that on her 3 year old.