Why do I suspect this is a "help me convince myself to take on a new car payment" thread? I once had a friend ask me if I thought his 5-year-old truck needed a new battery. I determined that it did and told him if he went and picked one up, I'd change it out for him. You can guess that he came back without the battery in a new truck. Whatever! Didn't take much to convince him! It would have been nice if he had at least offered to sell me the otherwise perfect truck for what the dealer gave him, though.
The reality is that the ice problem is your tires. I live next to a ski area, and the winding mountain road here has been solid white since October and will be until probably April. All of my vehicles have a set of snows (studded) because it makes all the difference. I even have a pair of studs that go on my utility trailer. Once the tires are changed out, life goes on without drama. Travel on ice in the winter is easier, faster, and safer than summer due to the absence of RVs and wildlife watchers!
You might not want to run studs because of the prohibitions in that region, but it sounds like you need something. The first rule of winter is that all-season tires suck on ice. The lower profile, performance-oriented tires, suck even more. I have never seen a vehicle with good three peak - snowflake, snow tires that couldn't function on ice and snow. My observation is that a vehicle that handles well in the dry, will handle well in winter, once equipped with proper tires. A vehicle that handles poorly in the dry, will handle poorly in winter, with the proper tires, but it will still stop, start and turn.
I own both types of vehicles. A FWD Lumina sedan from the prior century, and a newish AWD Fusion - with far too many electronics. Those are the good ones, and my winter weapons of choice. The old one with the big V6 that has torque and compression braking instantly on tap is more enjoyable to drive quickly in winter than the new 2.0 turbo with neither, despite fewer driving wheels. I have a 96 Grand Cherokee with about a 4" lift that's my go-to when the snow is getting more than a foot deep on the roads, but I have no desire to drive it in the winter unless I have to wade about in deep snow, or tow something. It just handles poorly year-round, but with good tires, traction isn't the issue. What is an issue is the lack of manners during an unplanned drift at speeds above about 35, maneuvers to avoid the odd deer or moose, or a good ice rut swap at speed -that kind of thing. While I find a little sideways twitch at highway speeds to be perhaps entertaining in the sedans, the same thing is terrifying to me in a poor handling, high CG vehicle! That's usually the type of vehicle spotted upside down in the median (normally with bad tires!) for a reason, I think.
Getting to your issue with ABS, I find the ABS in the 2019 Fusion to be very good. The 96 Jeep is also rather good. The 99 Lumina has a terrible ABS system and has scared the hell out of me under certain circumstances a few times. The thing is, ice isn't really a problem for them, especially the newer ones, as long as they have some traction to work with from appropriate tires.
Case in point, the Fusion got two new sets of tires and wheels to replace the stock 245 wide 19" low profiles as soon as I got it home. 225/60R16 Firestone all weather for the not-winter season (I still have to deal with snowy roads in the summer), and 215/60R16 Hankook iPikes, studded. The all weather tires are quite good in snow and slush, but as mentioned above, as an all weather {~season}, though these aren't awful, performance drops off quite a bit for ice. I had put them on for a long trip during a spring warm stretch, and I still had them on when the ice made a comeback. Pulling up to my cluster mailbox as I had without issue all winter, it was brakes denied! Same as your issue. Luckily, they did have just enough traction to steer around the mailbox. If my OE low profiles (M+S rated) could have made it out of the driveway (a maybe, due to the AWD), I would have surely killed the mailbox and car with those. I only keep them for the odd autocross.
So, I don't think it's an either/or question of snow tires vs. replacing the vehicle. It sounds like your conditions make it a matter of: You need snow tires, then the question is one of getting them for your current vehicle, or it's replacement. Problem solved either way, no big payment with the former.
And if it was me selecting a replacement, for pleasant winter driving, it would be nothing taller than a crossover, unless there was some need dictating something different. It's not the Tahoes and pickups that rule the winter rally races. It's the WRXs and the like! I'm not so narrow minded that I don't have a 4 Runner (that I won't lift!) on my watch list, though I will continue to own a sedan for as long as I can get them. AWD is a modest plus, if you really need it, but not a game changer for me to the point where I would reject a FWD if it otherwise made sense, and I get 16 feet of snow a year here. It's all about the tires!