Peter wrote:

You would however get into trouble if you recorded ATC and then openly
published the recording, e.g. on a website. I don't know why there is
this sensitivity because UK ATC are generally highly professional;

Maybe a parallel to Germany might help. In Germany, tapping into any
communication not intended by the sender to be heard/read/whatever by
oneself is illegal (regardless of whether there are measures taken to
prevent this) under privacy laws. There are some exceptions, but that's the
general rule. ATC communication is only intended for a limited circle of
recipients. Plane spotters are not the intended recipients, thus, they may
not listen.

It has nothing to do with perceived professionality of the ATC people, and
everything with protecting the privacy of ATC and pilots.