If an airport has winter weather infrastructure sufficient to handle only a small fraction of the flight schedule, how is it decided which flights get to go? Is it, like, the routes that "important people" tend to fly, like ATL-LGA or ATL-DCA? Or is it international flights (but would they leave half empty as connecting passengers wouldn't make the flight?)? Or some other criteria like planes that will have the greatest number of passengers on account of factors like the passengers are mainly locally originating and hence did not miss connections?

I'd love to get some perspective from someone who knows something about flight dispatching.