I can't tell you much about the car side of it. I CAN tell you a lot about the movie business side of it (a close family member is President of FX Network, I've spent time on sets, worked on several films, etc.)
1- It is unlikely that ad is from the producers, or the film company. It's from a contract movie car company who already has the contract for the show. They are your COMPETITION for the kind of work you are looking for.
2- Movie cars are about the contract, not the car. Producers don't care much at all about the cars. They care about working with 1 reliable company who can supply anything and everything they need, carry the insurances they need, handle the payroll, etc. etc. They DO NOT want to be in the business of finding 200 different individual car owners. Movie car companies do this, not Producers. THAT'S what they are paid for, not the cars.
3- Producers don't care about period correctness. It costs them $0 at the box office to have a piston powered car in a rotary. In fact, they usually MAKE money on small mistakes. Car guys get online and bitch without mercy, but it creates a buzz that sells more tickets at the box office.
4- There is ONE period correct issue they have- model year. They can't film a movie set in 1962 with a 1969 Mustang in the background. See #1- that's what movie car companies do.
5- It is necessary to have car doubles to make movies. Lots of them. Movies like F&F probably have 8-12 copies of EVERY car in the movie. But they are not identical- some may be just rollers, some may only be good for long distance shots, some may have a good left side but a crapped out right side. You HAVE to have full-time access to a fast paint shop that can make 8 cars look identical in about 4 hours flat, and rebuild a car when wrecked very quickly.
6- Movies shoot 24/7. They don't care about people's schedules, or why the car can't make it to the shot. You HAVE to be able to keep the car running 24/7.
7- Better be ready to work with Unions. The only people who drive are the Union drivers. If a shot involves an actor behind the wheel, OK. Actor drives 20'. But when the shot needs to be re-shot, the actor gets out, the Union Car Backer-Upper gets in, backs it up 20', and the actor gets back in to re-shoot the scene. No exceptions.
8- Production crews don't usually hire "car guys". They subcontract to movie car companies, who handle all of the above-mentioned details, logistics, and legals.
9- When a Producer says he wants 250 period cars, get them. When the Director says in the middle of the shot that he needs 50 more period correct cars, you'd better be able to come up with them in a couple of hours max.
10- Production companies burn a LOT of money. Many FX productions cost about $8 million PER DAY of shooting. If it is day and they need night, they filter the cameras. If it is night and they need day, they add enough lights. If it is dry and they need rain, they use fire hoses. They CAN NOT afford to wait on a car guy because he is having a problem. That's why they only contract with the big guys.
11- There are no full time jobs in film. It's all contract work. You are hired for 1 show, you work your ass off 24/7, then you are fired, and you start looking for the next gig. Producers have favorite subcontractors for every aspect of a production, and go to them regularly. But there are no staff positions. It's contract basis only- feast or famine. It also makes it a pretty closed shop- it is tremendously hard to break into a Producer's preferred subcontractor list, because he already has someone who can do absolutely everything he needs.
I did Special Effects on a couple of movies, and desperately wanted to continue (as a career). Even with my connections and on-set experience, it was really hard to break in.