Turning off auto squelch does not change anything. The noise appears the same way.
It could be atmospheric. It is not any kind of "machine" noise.
I was thinking I have a bad RF (antenna) connection but the really weird thing is that listening to distant transmissions when on the ground (with the engine not running - that is something I need to test) works perfectly. And the NAV channels both work equally, with the flag disappearing more or less together, which indicates it is not an RF signal loss.
Nobody I know of in the UK can test a KX165A from first principles. If it is that, I would have to buy an overhauled exchange unit from the USA.