My children are in their 30s now, and much has changed since then. London schools are a lot better than they were. However, I still think rather than demonising those who have done the best for their children however they could, we should all be pressuring politicians to improve state schools and to tackle the huge crisis in teacher retention. I don't just mean putting more money in.
When I was a child in the 60s and 70s teaching was a good, safe job with a decent salary and an excellent pension. Teachers were well-respected for the most part and had almost total autonomy in the classroom and very little scrutiny, so it was a high status job, albeit not one you went into to get rich. Put all of that together with the long holidays and it was a very attractive option for many graduates. A family with one full-time teacher parent and the other either a SAHM or (very often) a mother working part-time once her children were at school, increasing to full-time later on, could easily afford a nice house in a decent area and some sort of family holiday every year.
In the years since, teaching has become a far less attractive profession because salaries have fallen behind other professions, the pension scheme isn't what it was, early retirement is no longer an option for most, but also teachers have lost a great deal of that autonomy and now work under a huge amount of scrutiny. They are also not respected as they were. Both parents and children question thier authority (not all, but enough to make this a big problem). Discipline in some schools is also a big issue, which is another factor driving many teachers to leave the professional altogether.
I don't know how the government could solve all of this. They certainly can't do it quickly. But unless they try, that's what's causing the shortage of STEM and other teachers. The fact that it's generally easier teaching in private schools is not the main reason state schools have a shortage. People with STEM degrees can get far higher salaries in most other jobs and far better working conditions. The number who train as teachers and end up in private schools rather than state is a drop in the ocean.