I spent my entire career working for a natural gas pipeline company. My responsibility was to keep gas flowing throughout the system. I lived in a hurricane prone area and would watch as a hurricane entered the gulf and offshore wells shut down. Our pipeline pressure would go from 900 psi to as low as 200 psi! But never once did we lose our supply.
So yes the natural gas supply is vulnerable to disruption. We would see the same situation with winter storms. Natural gas (if available) would still be my preferred choice of fuel but I would have a diesel backup.
Another thing and I've never found this, would be a timer circuit so when power goes out and you're not home due to evacuations or vacations then the timer would only let the generator run for a couple hours a few times a day. This would be much better than it running continuously and would keep the freezer and fridge cold.