> because the ground infrastructure can't clear the snow quickly enough, or because the reduction in visibility means increasing separation requirements

Only true for airports in the middle of big flat nothing. If there're mountains around, bad weather or visibility makes landing too unsafe, unacceptably so for commercial airlines.

I live near this international airport: https://www.youtube.com/watch?v=TxzQIvVu7mg Closes in bad weather regularly. Doesn't work at all after sunset.