Part of the current "school run" problem actually goes back to the 1980s and 1990s, when parents were encouraged to seek out the best school for their little darlings, rather than simply send them to the local school. This meant that many students didn't have a direct bus to get to school . . . hence more parents drove them to school.
More recently, LTAs have scaled back their school bus networks, or even reduced them to "entitled" students only. In Hertfordshire, this happened in 2012, when the County closed all their school contracts, and left it up to "the market" to provide them.
I was at Mullany's Buses at the time, and we had around 20 contracts that would have been reduced to about 4 contracted school runs. We took the balance on commercially, on the understanding that fares would need to rise by about 50%-70% to meet the operational costs . . . we expected to do this over three years to try to keep the increases down. Some routes only lasted one year, as parents couldn't afford the increases . . . some lasted three years, but all went within five-six years.
The financials required a decker with a fully-seated load (so occasionally students would need to stand, which the parents wouldn't accept). We asked some schools to stagger start / finish times, so we could get two runs out of one bus and driver . . . they didn't even respond to us . . . those were the first routes to go.
One of the biggest problems is the demographics of students . . . we had a private contract that required two full deckers from NW London to the school in 2012 . . . within 10 years the numbers dwindled to a handful . . . now the remaining students have to come by car, as the financial case for the bus is no longer there.
The only solution is loads of minibuses on the "many to one" principle, but as the driver is the biggest cost by far, there's no way that parents will pay for that, and schools can't subsidise any network financially, so . . .