I think it's worth exploring options because this will come up again every school holiday and he needs a plan, definitely to cover the situation that she does need him or becomes not happy to be left, if he works a distance away.
It's easy for me because as I'm a teacher I'm normally off when my similar-age DD is off. But I have a few days in July to cover so I'm paying a teenage niece to be there those days.
I'd be wanting in his situation to have someone on a small retainer/wine/pay back in kind (friend's mum, neighbour, teenager, relative, cleaner/housekeeper) so there is someone about for some of the 12 hours if necessary. I'd also, frankly, be looking for a job change to something that allowed some WFH. I cannot imagine a mum doing this (short of emergency) and my goodness she'd get some shrift if she did.
If he can't afford to pay anyone anything...what did he do when she was younger?
However, I don't think it's reasonable to expect a 12 year old to take herself to "activities" she's not fussed about. That is unreasonable - why would she?! Besides that raises the issue of her not being at home but no-one actually knowing where she is, house not properly locked, etc.