Wouldn't work because you could complete the same trip via a range of different companies. So to keep track, you'd need to be tapping in/off each train so the operators could charge you accordingly. And then you'd never have an idea of what the price would be, as fares change according to how early you book and how busy the trains are.

I'm not saying it's a good system - it's a very bad one, having it split between so many companies is a mistake. But the London metro is not a good analogy since it is all one company.