Oh, that. Not with huge decreases in the birth rate that have been known about for years (multi authority panels have been operating in order to reduce overcapacity gradually so that fewer state schools close, but it's still seen a significant number do so). Covid and Brexit did speed that decline up, however, as so many people left - the cost of housing hasn't helped, either, as people have had to move considerable distances in order to afford somewhere to live.
There are some particular, massive, MATs who do seem to act as though they own the LAs, but they'd also be after the ex privately educated (on top of hoovering up as many as possible prior to census so they get more money - and then 'encouraging' others out to Home Ed before Spring Census with threats of penalty notices and prosecutions, but strangely, never wanting an attendance order or to find them suitable off site provision).
What seems to be a problem is where there are EHCP consultations and it's absolutely inappropriate for the LA to be pushing for a state funded mainstream school to admit. I've seen consultations where they've stipulated a daily session in the therapy pool with an onsite physiotherapist when the closest thing the school has to a pool is the largest pothole in the staff car park. They could have just gone with the superbly equipped special school that the child needs and has a place for them, but, no, mess the family around, try and persuade them that there's no right to Tribunal, threaten them with prosecution that they either go to mainstream or better still, EHE so there is no further duty towards the child.
Like I said, the key to standard admissions is not having all funding based upon one day in October. Being able to access money to pay for additional staff, meet K code needs, pay for school meals, instantly access high needs funding for EHCPs or people fluent in specific languages (including BSL), would create a huge incentive to admit in year and admit quickly whilst being able to meet the vast majority of needs, whether medical, physical, emotional or learning.
The other aspect is staffing and the huge amount of work behind the scenes that is necessary at the schools. LAs have dedicated staff working solely on admissions. Schools don't. Looking at that download or reading an email following FAP if they're lucky that might have the parent's contact details on it (but it's by no means guaranteed) is nowhere near all that is needed to get a bum on a seat.
If there is a school somewhere that can click a button and the record is generated, all syncs happen instantaneously, it talks to the catering payments system, it talks to FSM (when the LA access isn't given to schools), it talks to the canteen till system, it sends out emails, arranges appointments with a member of teaching staff on a period when they aren't teaching, doing cover instead of PP or dealing with an urgent safeguarding matter, finds DfE records of prior attainment, searches and finds a previous school, notifies them automatically on the first day, creates an individual timetable, sends details of uniform, data permissions, takes biometrics, performs CAT4 assessment and immediately updates 5 separate assessment portals and writes back with targets, prints the timetable, meets them on the first day, allocates a suitable buddy, orders additional stationery, provides access to multiple learning platforms and all the communications apps whilst also automatically putting in requests for safeguarding and SEND records from a school the parent hasn't even disclosed they attended previously, whilst also completing the shitty word document that should be a spreadsheet to a person two desks down at the LA to say they've been admitted and here are the details of the 300 others on the waiting list that now can't be offered a place, I'll be all for it. But there isn't.
These things take time - if all the stars align, yes, there will be a quick turnaround, usually because the majority of those things are all done by one person and they work flat out to get it done to the exclusion of everything else they do. But for the rest, it can't be guaranteed.