You may do fine and never have a problem, but the idea that fat sway bars hurt snow / ice performence has merit. Even stock smaller sway bars compromise performance in one area to enhance it in another.

Imagine two vehicles of the same weight and with a center of mass located exactly at the intersection of the two diaganols drawn from LR-RF and RR-LF so that each vehicle carried 25% of it's weight at each wheel normally.

Imagine that the one has the front and rear axles simply pivoted at the center with no springing to any wheel ...
... and the other with both axles mounted to a solid frame which allows no twisting.

On a perfectly flat plane, each carries 25% of it's weight on each wheel, perfect distribution.

Now place the same two vehicles on real world roadways where seldom is the travel surface all in the same plane. Now the one with the axles pivoting at center still carries it's weight evenly distributed amongst the 4 wheels ...
... while the other with solid mounted axles and stiff frame carries it's weight on the two opposite wheels (LR&RF or RR&LF) while the other two carry none.

This is why off road tractors have freely pivoting axles at one end to provide free articulation without unloading drive wheels. Adding thick sway bars and frame stiifeners is a shift towards that other with the solid mounted axles and rigid frame.

With a locker type differential, weight lost at one wheel on an axle is still lost and picked up by the other wheel, but the locker locks the two together so tractive effort is also transferred and you'll likely do OK. With an open type differential, the unloaded wheel will spin and no power is tractive effort is ttransferred to the wheel carrying that axle's load now.

I have two Thunderbirds, a '92 Sport 5.0 with shortened stiff sport springs and thick F&R sway bars and a '95 LX 4.6 with softer LX springs and thinner sway bars. Both ride on 16X7 wheels with 225/60-16 Goodyear Eagle RSA tires, roughly equal tread depth. Both are esentially, the same body / floor pan structure, unibody cars. Both cars have open rear ends.

The '95 handles OK but with more lean in hard turns, but it'll handle unever transitions without a hint of spin, it never while the '92 can hardly help but spin.

The '92 is great on smoother surfaces in corners but it's hard to leave a parking lot through a drainage transition or pull out of a u[phill side road through a transition to flat roadway without wheel spin, it's really bad if wet.

The '95 allows the tires to more freely folly the roadway irregularities, the '92 acts more like that rigid frame vehicle with solid axles.