I just went through this, for the 5th time, getting Dad's 36 De Soto Airflow to him.
It has turned into quite a racket, and quite frankly wore me out.
As you said, the enclosed is a lot more money now. I used "Horseless Carriage" back in the late 80s when I sold my 57 Fury, and again in 99 when I bought my Hemi RR.
But now, most of these companies are just brokers, using independent haulers. If you google auto transportation, you will find a site where you can get "multiple quotes."
Once you hit that send button, you are in for a nightmare, that will last about 2 months!
They will wear you out!
If you take the lowest bid, then the driver isn't going to be excited about taking your haul. Some of the brokers bid real low, just to get the contract, but then your car can set there a long time.
I ended up raising our offer by $75, and making the broker do the same. So we went from $650 to $725, but the drivers end went from $500, to $650. See the broker keeps your deposit, then the driver gets the cash on delivery for himself. Of course the broker did this to get me off his case.