I have a full winter on my Stealth P3D using Pirelli Sottozero II's and have driven my wife's LR RWD Model 3 on the OEM MMX 4's in the snow and ice a couple of times. I also love driving in winter conditions and have driven a wide range of FWD, RWD and AWD gas cars through winter storms (usually on curvy mountain roads but also across the prairies).
IMO, the Model 3 does very well in snow and ice although it does give the driver more leeway to put the power to the tires even when slip is detected. The "nanny" is very permissive.
That said, there is a sweet spot with the throttle modulation when it's really slick that will keep you well connected (no one else in the car will be able to detect any slip). You just need to drive like you are on a very slippery surface. The traction control will assume you want to maintain traction (and it's very fast and seamless about kicking in). It's only when you use the accelerator to tell the "nanny" you want to play that she gives you more leeway. I love it! But when driven gently, even on extremely slick surfaces, the nannies are super fast reacting, so fast you might think they are not even active. But as soon as you get a little frisky with the throttle she lets you play.
I think a contributing factor to some people claiming the nanny is too permissive is related to the way the tread blocks will wear on a car with as much effortless torque as the Model 3 has. If you use the accelerator much on wet or dry pavement even moderately (this is the equivalent of flooring it at high RPM's on ordinary cars) the tread blocks will wear into a sawtooth pattern. Especially when using winter tires with their softer rubber compounds. This sawtooth wear pattern on the tread blocks from acceleration creates what I refer to as "directional traction". Specifically, it reduces traction on snow and ice under acceleration and increases it under braking.
I installed winter tires on my P3D yesterday afternoon (the same set I used all last winter) and the sawtooth wear pattern was rather pronounced. I really don't mind this at all because the AWD of the P3D has all the climbing (and accelerating) ability I need in the snow and ice and it actually improves emergency stopping on snow and ice (and control under braking on snow and ice). I noticed this same effect on my FWD Volvo S80 T-6 where it was more problematic because that car was below average in snow/ice hill-climbing ability even though it was above average at cornering on snow/ice. But here's the thing - My AWD Mazda CX-5 never developed this sawtooth wear pattern because it only had 155 hp. Even though I often drove it semi-rally style (pedal to the metal) it simply didn't have the torque to the wheels to get a pronounced sawtooth treadwear pattern. It had pretty good nannies but they seemed overly intrusive when trying to drive in a sporting style. They were very competent but no fun at all.
So, yes, treadwear patterns can affect your perception of how well a car maintains traction under acceleration on snow and ice.I have played with "Chill Mode" and setting regen on "Low" when driving through the winter wonderland that is the North Cascades with all of it's grades and switchbacks but have settled on leaving it in standard accel and regen mode because I don't see any advantage to the "softer" modes. "Chill Mode" is particularly bad to use in snow and ice because it adds a small time delay and reduces repeatability of throttle application. I absolutely hate it.
I've found normal regen to have very fast-acting traction control (it cuts/reduces regen when slip is detected) but it still leaves enough regen braking that the tires can form a wall of snow or slush in front of them which increases braking under those conditions. When I add additional friction braking and more slip is detected, the regen braking backs off even more. If it's slippery enough the regen braking will back off completely in order that directional control can be maintained. So I have no issue driving down steep hills with full regen and simply modulating the regen with the throttle (and the directional nannies are kicking in as needed to help maintain directional control as well).
I find the Model 3 is the easiest and most relaxing car to drive fast in the snow and ice, uphill or down, without losing control or feeling like I'm fighting the nannies. Yes, it's a bit "looser" than any other nanny equipped car I've driven but it maintains directional control better even when it's loose. And I absolutely hate the abrupt way that most car nannies cut power the instant they detect a tiny bit of slip. The fast-acting but not overly invasive TC is why I think the Model 3 works better. This is comparing it every other car, truck and SUV I've driven hard in the snow including a long list of Volvo's and AWD Subarus, AWD Mazda, FWD VW's and AWD Ford pickups. I'm not counting all the cars that were pre-nanny. That said, the AWD and Performance Model 3's have different software from each other and I've only driven the Performance and RWD Models in the snow and ice. So maybe the AWD (non-perf) is somehow different. But I doubt it.