Why is it better to let your wheels roll in ice? I've recently read that if you're driving in icy conditions and begin to slip, it's better to not tap on the breaks, and instead, allow the vehicle's wheels to continue rolling. Why is this the case? It doesn't seem very intuitive. I think it probably has to do with static and kinetic friction, but I can't put it together.
 A: I'm not entirely sure why they would think that.
Tapping the brakes is almost definitely going to be better than not tapping the brakes. 
 Using the brakes causes the static friction between your tires and the road to increase; which slows down the car more.  
The issue is that when the grip between the brakes and the tire is too strong, the coefficient of friction between the tires isn't high enough to maintain that static contact.  This is when the wheels "lock up" and begin to skid.  As you mention, kinetic friction is less than static friction, so wheel lock and skidding should be avoided.
This means that tapping the brakes is generally the best braking method in these conditions, as compared to not braking at all.
When considering modern vehicles, your friend missed the mark even more.  Most vehicles sold now (as far as I know) have an ABS, an Anti-lock Braking System which actually disengages the brakes when they reach the threshold when you start to lose traction.  Basically, you can push down on the brake pedal as hard as you want, and you can feel the brakes start "tapping" on their own, to prevent skidding, increase traction, and allow you to brake in the lowest space possible.  Tapping is the method most cars automatically employ on icy surfaces, and if you try it for yourself (in a controlled environment), you will find that tapping the brakes should slow you down a lot faster than not pressing it at all; especially if you avoid skidding.
A: You're right, it does deal with static and kinetic friction. We know that in this situation (whether we tap the brakes or not), the frictional force is acting opposite to the car's motion. Let's consider both scenarios:
Applying brakes: When we apply the brakes, the wheels are "locked up", so they are simply being "dragged", therefore the wheels are experiencing a kinetic frictional force. The net frictional force (acting against the car's direction of motion) can be calculated using F = N×μk, where N is the normal force of of the car (N = m×g), and μk is the coefficient of kinetic friction. 
Not braking: When we don't apply brakes, the wheels are actually experiencing static friction. This is because at any given instant when the car is moving, the  whole wheel is actually rotating around a fixed point, which is the point of contact between the wheel and the ground. So, at that instantaneous moment, the point of contact is not moving, and therefore doesn't experience kinetic friction, only static friction. Similar to kinetic friction, the formula for static friction is F = N×μs, where the value of μs is greater than the value of μk. 
So, the static friction force is greater than kinetic friction force, which means that if the tires are rolling, there will be more frictional force, so greater deceleration, meaning quicker slowdown. 
