A real-world (and easy) explanation for the problem are your tires... R4s have a very hard rubber compound and have no siping whatsoever. What's siping? It's the little slits that you'll find in the tread pattern on certain tires... BF Goodrich Mud Terrains? No! They're just like R4s - designed for off-road, rocky, gnarly ground that would cut right through a car tire. Nokia Hakkapeliittas? Yes! They're dedicated snow tires with thousands and thousands of these little 'slits' in the tread pattern - and the rubber compound is very soft so they can flex. They're designed to fill up with snow as the tread closes up (as the tire rotates) and literally "stick" to the ground on the next rotation. Here's a picture:
http://www.canadiandriver.com/articles/hl/images/hakka_5-1.jpg
Your average "mud and snow" tire is somewhere in between - because those 'sipes' make a ton of noise as you're driving along and would make most people crazy. Also, in the summer, the tread compound is so soft, you'd wear them out in no time.
Now, I fully realize that we don't put snow tires on a tractor - but the lack of siping on the R4s prevent the tire from having any traction to speak of. The municipalities here in Vermont actually run different sets of tires during the winter - smaller turf tires for the John Deeres that plow the sidewalks. No chains - but no hills either. The turfs give quite a bit of traction compared to the R4s. But, the reality is that on a slope that's coated in either ice or hardpack snow, there isnt a tire out there (studded, I suppose would be an exception) that's going to give you traction to stop.
Two things you'll find with chains - (1) you're going to be in for sticker shock if you get a proper set of chains for the R4s and (2) they're a royal pain to put on the first time, then only mildly difficult each season afterward. Traditional "ladder" style chains won't work because the links will fall into the tread - that is, unless you get such a large chain link size so there's some metal exposed beyond the tread. The set I bought for my 2320 were $400, discounted pretty well and have a 'cross' pattern that leaves the chain on top of the tread, rather than between it. They're heavy as can be. But reasonably easy to install - place them over the tire (yes, that fender gets in the way!) and drive forward slowly until the ends are close together. But the first time you go to install, you've got to put them on for measurement purposes only - then remove them to cut the extra links out. Mine are 3/8" hardened links and certainly can't be cut with a Sawzall or hack-saw. I had to take them to a machine shop and get them cut with acetylene. I can now go right over ice - and only lose traction when I've pushed so much snow that it overpowers the tractor.
If I had an experience like your's, I'd be looking into a set of front chains, too.