I think I'd poin out a few things here:
a) Nothing wrong with comparing with the private sector, but take into account all benefits also. I guess most private sector teaching is private schools so they will be paid better, but whether you compare to other jobs like in Finance someone said, you'd have to compare giving up the holidays and copper plated pension.
b) Schools are becoming academies now. At some point pay will start moving in a more private sector direction. So you'll get what you want to some extent, but there are good and bad points.
c) Those that work in public sector automatically get pay rises to the next level, especially in teaching, and they usually exceed the rate of inflation. This is not always the case in the private sector. However, in the private sector, what happens is that people move jobs and then the role is re-evaluated as to whether it is competitive on the market when no one applies.
d) Private sector can be a bit weird when there are huge fluctuations. E.g. someone could get employed on X per year and because we have lock down and people move into other industries upon recovery and then there is Brexit and there is a lesser pool of talent, the rate can shoot up, someone gets in on X+10 and then companies feel poorer, the market calms down and then the rate is X-5. People moving jobs / made redundant get caught up in this storm for better or for worse.
But to get to what it would be like if schools worked completely like private. For the most part the academies would solidify like the supermarkets and in general pay would yo-yo like in the private sector where you wouldn't be banded but there are simply roles to be fulfilled and different people doing the same job would be on wildly different pay. In the teacher pay region you'd be looking at a variance of for the most part £10K. The ring-fenced annual increment would likely be below inflation and you'd be looking to move jobs every 3-5 years (there are exceptions like if you landed on a high pay, or if you change roles within the academy). You wouldn't get pushed up bands automatically.
Work would also change so that by and large you'd have more flexible part time and full time roles. Different groups would be doing things like marking homework. The concept of a long summer holiday would probably disappear. Teachers would rotate / more teachers in a class to cover each other. The teaching roles would have other admin / planning jobs over the summer and your time would be cut down to 4-5 weeks per year, but you'd be able to table them outside of the expensive summer for a good chunk of them. When budgets are not being met then you'd be made redundant if your role is not required - this would have nothing to do with how good you are - unless you have other peers fulfilling a very similar role at the time.
TL;DR when you ask why jobs cannot pay the same as the private sector, you need to understand what you're giving up should that happen. Whether public or private, there is a budget, so if you get an increase, someone else gets decrease is pretty much the bottom line.