And if the dad was disabled in such a way that doing the school run made his disability worse, made him less able to be a SAHP, and risked him having a meltdown and losing the ability to speak or make safety judgements whilst supervising the children in the presence of vehicular traffic?
None of this is what OP has actually said... so you seem to be making up a scenario and catastrophizing to make a point, when none of that has come from OP? She said it stressed her out, gave her anxiety, it felt like too much, and she is happier to not have to do it now, end of.
But if OP can get to her volunteering role (without panicking about being late, or writing things down 5 times, and without being distant and stressed out with DH and the kids every morning and afternoon), navigating public transport, volunteering inside a busy hospital, volunteer in foodbank, navigate crowded places and busy transport systems and it doesn't bother her in the slightest, I wonder how much of the school run issue is actually effected by OPs disability and how much of it is OPs fear of school which goes back to her childhood and is a known trigger point for her anxiety?
I don't disbelieve her in the slightest, and can fully understand why the thought of doing it twice a day every day would have her spiralling, but imo it seems implausible that OP can do everything that she does outside of the school run without batting an eyelid, and actually says she really enjoys it and it doesn't stress her in the slightest, yet leaves the kids in afterschool club for 2 hours twice a week because she can't cope with picking them up.