The moment "s/he" mentioned a "pilot," that's all the evidence you need to stop listening.
But, will the fridge burner go out while driving? Probably not, but it's easy to check.
1. Use the belt and suspenders technique the first couple of times. Freeze some blocks of ice or some normally unfrozen items. A gallon of milk freezes nicely. Just be sure to pour off a bit to allow the milk to expand when it freezes. That's your backup to keep things cold just in case.
2. With the fridge at operating temp, and with it on propane for at least a few hours before departure ("force" it to run on gas even if you're plugged in to shore power), check the fridge for a "fault" or "check" light. Mine also beeps at me if it stops running. All good? Then ready to roll.
3. Now head out with the rig in tow...fridge running on gas. After about an hour, pull over and do normal "in-flight" inspections. Tires, loose stuff, hitch and the fridge. If the fridge has a problem, the fault light will come on. If not, the panel will display as normal...same as at home in the driveway.
4. Going a long distance? Well, you've gotta pee sometime...and get fuel. Check the fridge, tires, and the rest again at every stop. These are normal towing checks...just add the fridge to the list.
More than likely, all will be well. If not, your suspenders will keep the contents cold.
If the fridge faults on gas on the road, when you come home, pop the lid on the burner cover and explore options to create a wind break to shield the burner. Of course, you MUST leave plenty of air flow, but even something as basic as metal window screen could admit plenty of air, but create a bit of shielding from the slipstream while pounding down the road. Bonus points on the window screen: no mud daubers or other critters moving into the fridge burner area.
As for the dolt talking about pilot lights, you need to double-check EVERYTHING this person told you.
That was lame.
Good luck.