Also: UK ETA can't be used to live nor work in UK. Dixit UK gov.

Only existing life choices protected under EU law were included in WA. See below excerpt from De Rynck's book for an example of an EU right that was lost if it had not been explicitly used already: family reunion. I don't think there is a section on part-year residency rights in the treaties.

TLDR: residence seems to refer to the State where one habitually resides. If one resides in more than one State, only one State is the habitual residence. In terms of exercising residence rights, discussions seem to revolve around determining which State that is.

www.law.kuleuven.be/iir/nl/onder...

This is from europa.eu. It clearly says "in your host country", not "in one of your host countries".

Ok. You win. Let's look at the Labor Particpation Rate. UK 78.3, US 62.5.