The loading and unloading of the car is actually very straightforward. Follow proper securing instructions for the type of trailer that you have and this part of the process can be completely forgotten while on the road. Problems arise because not many people have experience driving large vehicles or pulling a trailer. Even a one car trailer makes every driving maneuver more difficult. Backing up and turning around are extremely difficult and should be avoided, if possible, by first timers. But things like switching lanes, making turns, and even getting passed on the highway are much different also. During a long trip you can completely forget that you have a trailer behind you. Usually this only results in running over a couple curbs, but switching lanes without thinking could be disastrous. Another common issue people have is underestimating (or forgetting) the height of the truck. If you consider yourself a distracted driver you will most likely be better off having someone else drive or consider other options.
Insurance can be another disincentive. Your car insurance may or may not cover the car while you tow it so explore coverage by the rental facility. You can insure your household goods and your car but only up to a certain amount and many types of accidents are exempt from coverage so do thorough research. If you don’t get any type of insurance you could be looking at replacing your items, your car, and the rental truck and trailer out of pocket.
Pulling a car yourself will, at the very least, help you empathize with auto transporters. When you try backing up with a trailer for the first time you will be in awe of someone able to do it with a trailer five times as long. Make sure to square away insurance well in advance and to research correct loading, unloading, and operation of the truck and trailer. If you are not completely confident driving a car or if you have a history of poor driving, do not even consider this auto transport option. You might be able to get away with driving the moving truck but you will have problems pulling the trailer.