Having worked in Austria in the past and using full winter tyres (Goodyear) I would just comment on the amazing superiority of them in snow/ice.
As you say, there are also the legal aspects in various countries, so you have to comply with some sort of solution. Germany is very strict.
I think if doing it nowadays and travelling to/from the UK, I'd be choosing a cross-climate tyre (I believe the deciding factor is whether the tyre is marked with a winter snowflake/M&S ). If living in those climates I'd probably do like the locals and have a spare set of full winters on steel wheels. Cross-climates, providing they comply legally, will certainly give good snow/cold performance and as long as sensible caution is applied I'm sure they'd do fine.
There are plenty of reviews and several top performing tyres to choose from now.
Chains are usually only permitted on specified high mountain roads, socks are a waste of time IMO. Too much faff unless you are seriously going into the wilds. Normal roads just need winters or C-C's.