A raw root vegetable can sit at room temperature or slightly cooler for weeks and months without bacterial growth that will harm you. The same is not true of meat, cooked vegetables, and other things you're likely to have in your freezer. That's where the "don't refreeze" advice comes from - the food may have started to spoil, and you should therefore cook it and eat it quickly before it gets worse. Entirely different from whether a raw root vegetable happened to freeze or almost freeze.
Some vegetables are so much nicer after a frost that my parents had a rule that if you harvested them earlier (because you couldn't wait to eat them or there hadn't been a frost by a meal that called for them), give them a little time in the freezer. Parsnips, for example, are much sweeter after a frost.