Car cover will flap in the wind and cause more damage.
It is pretty hit and miss really, I have shipped five cars cross country on open trailers and four came out fine. One they used a tow hook to secure the car but it wasn’t designed for that and had to be fixed. Two more cars delivered off Vroom both were good. Just depends on the care the driver takes, weather, and if they are behind schedule and rushing. If your car is loaded up front it won’t be moved around.
Clean and detailed before it goes on so you get good pictures. Take your time when they inspect on and off. Damage found later won’t get covered. Drive it around the lot before the truck leaves. Tip the driver 1/2 up front. Recovering any damage will be easier with a relocation company intermediary, but they contract with lots of companies and brokers.
My gtv6 was shipped recently in a closed trailer Cross country that is the way to go. Offer to pay the difference.
If you were considering PPF or a ceramic coating, now is the time but at least put a good sealant coat on if possible.