I had presumed that car GUI/Infotainment systems were just in sleep or hibernated when the car is off, to shorten the startup time when switched back on.
Some part of it, yes. In the systems I worked on you had the "AC" - Automotive controller that was in a sort of deep sleep. But the "CG" - graphics controller was completely shut down.
cheap electronics will stand up to use in a car for years - something the fastest computers often fail at. I'm under NDA so I can't give more details.
Those cheap often boot instantly while more powerful ones often need a minute by the time everything is initialized.