In every subject there is a shortage in secondary. Primary too in cities.
This is not true.
In English for example there have always been more applicants than jobs.
And looking at teacher training, the good unis are still very selective about who they take for PGCE courses.
If there was a major shortage this would be reflected in admissions and lowering the admission grades.
The issue with teaching and recruitment is that the starting salary is C£25K. This is on a par with a lot of graduate roles- certainly in DH's multinational company that is the norm.
The problems with teacher recruitment are complex.
On the one hand, you have grads going into teaching because they don't know what else to do. (This was always the case and from my old grammar school around 30-40% of my peers went into teaching as they didn't know what else to do.)
Then when they are at the chalk face they find they don't have the personality to cope with it- mainly discipline issues.
So they leave.
Or, they find that although the starting salary is ok, the opportunities for promotion are few and far between unless they want to relocate (which is harder if you have a partner in a good job or children at good schools.)
In my personal experience, SLT consisted mainly of useless teachers who wanted to teach less and do admin type roles. They were often promoted to SLT as a sideways step and sat there for years on good salaries.
If you compare the salaries of teachers though for the weeks they work each year, it's not bad pay. No other profession gets that much paid holiday ( it's what- 196 days a year of work?)
I know teachers do courses and prep in their holidays (got that T shirt, thanks) but on the whole they do work fewer days.