It is a severe crisis and disgraceful that things have come to this. This is just one visible problem, but there is a massive ongoing recruitment crisis under the radar. My niece in a London comprehensive has only ever been taught computer science by a various supply teachers since September last year.
About this though -
We really are a two tier society now, with students in private schools in small classes, taught by specialist teachers, and students in state schools effectively teaching themselves at a computer
I am not so sure.
From my personal experience, the crisis is affecting the independent sector acutely too, at least in London.
The worst such example I know of is when South Hampstead High School (quite a high profile school in London, 21K fees) jointily advertised for a chemistry teacher position with another nearby state school because neither could fill that position independently. South Hampstead had that position open for five months!
www.standard.co.uk/news/education/chemistry-teacher-london-recruitment-crisis-state-schools-b1057751.html
In my DD's senior school in London (about 22K fee), a long standing English teacher left, and an entire term went by before they could find a replacement. We now really worry about who would leave next!
I think teacher salaries are simply not competitive enough in this country, and as a result, many are leaving the profession for better opportunities elsewhere, fewer joining the profession, shrinking the pool of available teachers for both independent and state sector.
Just as you would expect in a free market.
To attract people more people into the profession, better conditions and pay are required.