With passenger cars it's not so much being high-centered, but rather snow building up under the front bumper and radiator, eventually forcing the radiator off of its mounts or worse. Also, the front wheel wells can really build up with snow obstructing your ability to turn the wheels; don't ask how I know.
My '98 Honda Civic was fine on highways during winter (S. Lake Tahoe, North Shore, Yosemite, Mammoth) with all season tires and cables, traction wise, but ground clearance did come into play on smaller less-maintained roads and parking lots. Bring a shovel.
My '07 Subaru Impreza, with dedicated snow tires (Bridgestone Blizzaks), has been the most stable vehicle I've ever driven on snow. This includes a comparison to my '03 Toyota Tacoma 4WD w/ dedicated winter tires (Mich X-Ice).

