65mph on the highway in snow cover or slush is way too damn fast, its quite easy to lost control and spin out. If you're driving nearly that fast, about the most you can move the steering wheel from dead center is 25 degrees either left or right. You must wait until the car decided to catch a grip and move over, otherwise if you steer even 1/2 of 90 (like 45 degrees) because of non response the car is going to catch and over-steer flying off the road.
That's the difference between all season tires and real snow tires in snow/slush/freezing/ice/etc.
With all seasons you just slip all over. With real snow tires you can go drive along much closer to normal.
Don't get me wrong, you still need to use caution with your steering inputs as you suggest, but there's a big difference between what happens when you make those steering changes.
Heck, if we're just talking about 30mph city streets then I wouldn't really worry much about tires anyway. Your biggest problem is just going to be getting stuck in a slushy parking spot.