Having spent 45 years in the trucking business, I think you have good reason to be concerned. The problem with any national transport company, is you are going to have a wide range of competence within their driver pool, even if they are pretty good, reputable carriers. With cargo that requires special handling, like vehicles or household goods (which I hauled during the '70s), the individual driver makes all the difference. Small, local guy with good references may be a better bet.
I had a '66 Corvette shipped from Atlanta to State College 12 years ago, and then back when I sold it two years later, and it was a fiasco both ways. I didn't set it up, dealer I bought car from went for the cheapest bidder. You get what you pay for and less. Horseless Carriage has been around forever and hauled for many high end customers. But in today's driver market, you might still have a problem.