When British Rail was created, what were the reasons for not making the
London Underground part of it?

Doesn't the existence of two very separate railway networks in London make
travelling in or across London harder (when separate fares and/or tickets
are needed), less well informed (due to relative lack of public knowledge of
the ability to make many journeys by NR instead of/as well as by LU, or vice
versa), and more expensive (due to missed economies of scale in management,
staffing and many other areas) than could be the case with one merged
network? What mitigating circumstances are there?

A metro system that is part of the National Rail network seems to work
perfectly well in Liverpool. Are there reasons why it wouldn't in London?

(Genuine questions from a puzzled non-expert.)