I think as teachers we really need to talk to support staff about the reasons for the school strike, and why they should support it, and at least not take on additional duties on that day.
It is also worth reminding support staff in an LA school there is a legal duty that a class is under the supervision of a qualified teacher.
BTW, support staff can refuse to cross any picket lines at a school on the day, even if their union hadn't balloted them.
I do think Unison etc should do more to advise their staff.
Obviously support staff are free to make their own choice, but I do think if the head asks them will they do xyz on a strike day, it would be good if they didn't say "yes, yes, yes" immediately. It's reasonable, I think, to ask them to say they will check with their own union first.
But it's worth reminding heads that if the school is picketed, any of the workforce can choose not to cross the picket on the day, so they need to consider the safety of all students, and whether they are 100% sure they will be able to keep them safe on the day.