I will relay my one experience. This was in 2019 when we made a move from the Midwest to our new home in Florida, a 1,200 mile move. I have a classic car that I needed to transport to the new home because I did not feel comfortable driving a 55 year old car 1,200 miles. Keep in mind that the car had not been driven a total of 500 miles in the previous 12 years.
We contacted a "broker" that specializes in auto transport. We got a quote and agreed to the price. We advised the date that the car would need to be picked up, and we were told "no problem". Well, as the day approached we called the "broker" to confirm the pick up date, and again were told "don't worry, we will be there".
The day before the agreed upon pick up date, I called to insure everything was on schedule. I was told the "broker" was out of the office. I called back several times that day to confirm they were coming but, of course, the "broker" was never available to take my call. So, you probably guessed it, they never showed. Therefore, the next day I was forced to drive the car all the way to Florida. They called a few days after I arrived in Florida to inquire if I still needed the car transported. I wasn't real polite.
My thoughts and advice:
Do not use a auto transport "broker" as they do not own the trucks, they only try to coordinate shipments as capacity is available from true auto transport companies.
Keep in mind that an auto transport will likely load/unload numerous times in a cross country move. Protect yourself by taking multiple clear photos showing the exact condition of your car before it is loaded. If there is damage in transit you want to be able to document it.
Be careful to use a reputable transport company because some will quote a price, show up with your car, then tell you they need a few hundred more $ or they will not deliver your car. This is a real problem. I am a "car guy" and know many people like me. I have heard the horror stories from actual "victims" of these transport companies.
Car transport companies will often not deliver the car to your new address. They will often want to unload in a large parking lot somewhere near your new home. You might have to find a ride to that location to pick up your car.
If you are moving your possessions via a large mover like Atlas, etc, they often can load your car in those big trucks right along with your possessions. The cost might be slightly more but it will arrive at your location with your other possessions.