I’ve not really had a mechanical related problem riding in the cold. I’d suggest that cleaning the bike may well introduce water into voids/ spaces/ interstecies in cables, mechanisms etc that will then freeze. I find that my brakes make a lot of noise in very cold conditions. And air seals leak more than usual- I’m assuming that they stiffen and thus become less flexible and resilient in the cold. My shock pump has had something of a workout recently.
Decent gloves really help, forget waterproof, you need to let moisure escape so that you don’t get sweaty and thus chilled. I use a pair of PI gloves, bulky but nice and warm down into sub double digits. Mits are even warmer.
In really cold weather I don’t use my winter boots (too constricting), I use my summer ones, with a pair of wooly boolies, another pair of merino socks and then loosly laced up, so that my feet are not squashed. Cold at first, warm as toast later although it can come in waves.
Roubaix bibtights, perhaps with Ron Hills over the top and if an icy ride is anticipated, knee-shin pads for warmth/ knock protection. Toasty until you remove them.
As for the bloke riding with bare skin- rather you than me. I came into this game via Hilwalking/ climbing/ mountaineering and would never leave areas of skin exposed in such a way in the winter. You may not feel the cold as much on your legs but it is still capable of doing damage.