A possible brokerage opportunity would be to inquire with Carvana or Carmax if you can buy slots on their cross country shipments as capacity allows, and sell those to customers. You'll be at the mercy of their schedules and capacity, but could be a service tier along with other more "white glove" services (dedicated driver "hot shot", semi shipment, etc). You could also buy capacity on freight rail cars shipping back empty to factories (depending on destination and routing). High touch biz, beware, there be dragons.
I think the fact of the matter is, shipping cars is not going to be cost effective for an individual. Trucks are expensive, putting fuel in them is expensive, paying a trained person to drive one is expensive.
Maybe a company like Carvana could offer this as a side service, drop your car off at a local Carvana, and eventually pick it up at another location. Even then, I think the costs are going to be order of thousands of dollars.
Just need to use a direct carrier like Intercity, NOT a broker. Brokers have infested Google SEO.
So because you mentioned it I just called them for transporting by enclosed carrier on the east coast between 2 cities. High end car enclosed carrier.
Intercity wanted $1995 for the same transport that the car dealer (a high end car dealer well established no less) want $1600 dollars for.
Intercity essentially said 'busy time of year etc etc'. Didn't give remarkable reasons why they would be better than anyone else. Picked up the phone right away and was otherwise business like.
https://intercitylines.com/request-a-quote/
Also their website to get a quote they want an 'auction' number. That made me call them (which I would normally never do.)
Additionally their price was higher than other brokers I had checked including the one that just moved a car for me (same route) not enclosed who quoted both enclosed and not enclosed.
Would like to point out that the job of a broker is not to match you with anyone but match you with people who they know and trust and have used. And specifically the broker I used (for the last open move) made a total of about $250 the rest paid to the driver on delivery.
As a consumer I'd assume there are hotshot truckers/services that would be used for this? People have mentioned similar "agent oriented" experiences with moving services and how it's opaque and unreliable.
The ones carrying cars across the country for moves are quite rare, unscheduled, and harder to find. They also can't carry anything else besides cars on the transporters - you might almost have better luck driving your car into the back of a box semi and letting them load other LTL loads behind it.
My local dealer somehow got a lead that I want a very specific year and trim GTI and twice now they have offered to bring one in for me.
You place anything you need shipped on there and various shippers can bid without having to go through a broker.