Some people do not want chemicals on their lawn. I am curious to know if freezing would be the "organic" approach.
Crabgrass dies at frost at least partly because it's genetically programmed to die every Fall. As you are thinking, it's also likely not adapted to be able to freeze and still survive. A frost might be simply burning off the top growth, but none will ever grow back because the plant is already super stressed by Fall and colder temps /shorter days generally. It's in its death throes by the time of frost.
Freeze spray is way colder than just freezing. I'd bet even many plants adapted to freezing temps would likely die if shot with freeze spray. Only way to find out would be try it. A can of freeze spray isn't expensive.
If it works, a several gallon tank of liquid carbon dioxide doesn't cost that much either. Get a tank with a dip tube and a valve on a high pressure whip so you can spray the liquid on the plants to kill.
If you can freeze spray the growing point of the plant, It should work just like gly.