Part time doesn't mean school hours friendly, and doesn't mean that they can accommodate a schedule which chops and changes for weeks on end.
Low income families are less likely to be able to afford ad-hoc short term childcare, too.
The low median income isn't just caused by lots of people in nice jobs working part time. Full time minimum wage is a long way shy of £19,000, and there a lot more of those.
We're nw England, and not the lovely Cheshire bits. DD school isn't full of lawyers/doctors/managers kids with parents panicking about meetings and deadlines. It's predominantly shift workers terrified that asking for half a term of less hours would mean they never got those hours back.
My family is one of only five in her class with either a sahm/part time parent or one in a professional role. That's going to get worse as we continue to switch to UC. We are lucky; I earn well working in an office and we've managed to swing DH only working 8-2 since DD started reception.
Before that, she was in full time childcare.
60% of her year came from the school nursery, which did a full school day. A good number of others came from other childcare. Vanishingly few had never left a parent.
Im not denying that staggered starts will be better for some children, but they're also detrimental to others. If an individual child needs a staggered start, they should have it, just as late born children should be able to delay starting at all.
But as a default norm, it's unfair to as many kids as it suits and hellish for families just trying to keep their heads above water. What's wrong with the approach taken by DD head?