Amused by your use of two more terms that have different meanings in the UK (although young people are confusing things by adopting the American meanings)!

The point is that British universities are almost entirely private institutions that get a lot government funding, in particular for British students (so they pay much lower tuition fees) and research.

There is a heavy reliance on government funding at British universities so they are probably to some extent comparable to American public universities. On the other hand some have substantial resources of their own. They can also turn down government funding, and some have threatened too at times when unhappy with the terms that come with it.

Given that Yale seems to have had a serious funding gap last year because the government reduced funding it seems comparable to what your sources call a "public" university in the UK

> Either the UK experience is with private schools, in which case the statement is irrelevant, or it is with public schools, in which case the statement is either wrong or irrelevant.

Yale has far more money than any British university so regardless of how you classify them, the question of why Yale charges for things British universities do not is relevant regardless of how you classify them.