Let me address some of the points you raised about public education. I think the choices here are much more about political ideology than about homogenity and race.

For instance the US setup a public school system long before Britain despite being a more diverse country. Germany did as well which got united into one country very late and thus cannot be called very homogenous.

What sticks out to me as a far more prominent differentiator is political ideology. Britain is the torch carrier in Europe for free wheeling capitalism and a minimalst state. The anti-government view of Americans don't come out of thin air. It has strong connections to British culture and society which is among the most negative to government in all of Europe.

The British did not want to create a school system for the common people because they were anti-government. They pushed the classic libertarian ideas of minimalist government.

The fact that Britain eventually got public schools was because they started falling behind other European nations. They noticed how German industry was getting far more technically advanced than British industry. Much of that was due to much stronger school system in Germany.

Even then the British would not really accept a public school system. They ended up with a very divided school system where the upper classes went to a different type of school than everybody else.

So in my mind it is perhaps not so much that Nordics were that profoundly homogenous. But rather that the British have long has a very strong capitalism/minimalist state oriented political ideology.