P.Pilcher.
Your reasoning is full of holes. My wife is a teacher and she and I both know that teachers are, in fact, fairly well paid for what they do. The problem is the paperwork overload and the constantly changing demands being thrust on them by whoever happens to be in power at the time. The fact that there are three teaching unions has nothing whatsoever to do with teachers levels of pay. This is because the money available for teachers pay is determined by the exchequer who can only pay what he can spare out of the national tax take.
Pilots, on the other hand, can nowadays only demand what they feel is due to them in preference to all the other demands of the commercial business in which they are operating. But, again, I don't actually feel that unions help them a lot either. The real golden age for British pilots was when BA was owned by the UK Government, and was effectively a monopoly. The largese and extravagance of BA around that time was legendary and the pilots pay deals were fantastic.
I personally am self employed. So having absolutely no-one else to whine to I have to earn it for myself. And before you start on about me being a 'fat cat', I can tell you that I earn rather less than the average teacher.