From New England where you have to shovel yourself out and get into work anyways so I appreciate that we rarely get snow in winter where I am in the UK, and when it does happen, it's almost always just a light dusting.
That being said, I MUCH prefer the warm. For me, it's less about the cold itself, but the cold accompanied by darkness when I leave the office at 5:15PM and dampness. I wouldn't mind it if I worked from home all the time and could choose when I wanted to go out into it, but I have a dog so can't just decide I don't want to walk her on a given morning/evening when it's exceptionally cold. The daily pains, IMO, aren't exactly miles away from what I dealt with in the US (E.G. defrosting the car/scraping the car off in the morning) minus the snow problem....but other parts of the UK will have to deal with snow far more than I would.
Can I tolerate the winters here? Sure. Do I actively enjoy them?: Nope. I almost definitely have SAD, so the brighter it is, the happier and more positive I will be and the warmer it is, the more I want to leave my house (which, again, has an impact on my mental health). I don't think it's necessarily down to location, however. I think some people are winter people and some people just aren't. Even back in the US, I absolutely hated when it would snow because it meant I had to go out into the freezing weather and shovel for what felt like ages. Some people love the cold and don't mind that but I would dread it. The less amount of time I needed to give in the morning before going to work, the better. While I enjoy some aspects of winter, I am just not a winter person regardless of where I'm located. (Sorry for the ramble, I've not finished my coffee yet so my brain isn't awake).
I think your experience with winter climates is very relatable. Any weather condition that requires additional preparation and is time consuming while feeling physically uncomfortable is bound to be more challenging. I have lived in the Northeast region in the States and had to navigate through a few crazy winters but they were never that bothersome. At least that's how it felt at the time. But you just get on with it. The Northeast is known for northeasterners and when one of those buggers comes through deep in the heart of winter you pretty much have a white-out condition. Three feet snow falls happened several times while living up there but even that didn't feel so bad. Anything over 2 1/2 feet would pretty much shut down the schools and several businesses for at least one day maybe two but then everything would come back online after that because they were brilliant at snow removal. After a while you'd get use to those conditions. Your blood would thicken up a bit and the cold didn't feel bad. A nice parka with a hoodie and you were off and running.
But my one year in Seattle with the high temp of 30-40 degrees and the damp windy winter weather, I felt exposed! The skies were grey even in July. That was truly shocking. And by November my fight or flight senses had me looking up driving routes back to the east coast. I told myself that this was a momentary thing and I needed to stay put, after-all I had tougher experiences living in NYC. New York is suppose to make you tough, you know the song. You can make it there you can make...……. Ok, that ain't true. Because when January hit I was 100% sure the Pacific northwest wasn't the best place to live in winter. Beautiful looking place and maybe one of the most scenic places on earth but those winters are too intense. As a side note, when I did move back to the east coast in march, I couldn't drive back the same route because it had frozen over. Montana, Wyoming ect were no go areas in the dead of winter. Instead I drove down to Los Angeles and crossed over. In hindsight very glad I had the experience I can check it off my list of things to try.