So yeah I think the future of CarPlay or Apple car is actually Apple hardware and software built in to other manufacturers cars.
It could also be a subscription - "CarPlay+".
In before times, BMW and others tried this model with the original CarPlay itself but eventually got shamed out of it. The backlash was largely because a major motivation behind the original CarPlay was the claim that it would increase driver safety. Indeed the entire point was (and is) to be able to use your iPhone's native media / navigation / SMS / handset software - all things that you'd be inclined to do while in your car - without actually physically interacting with it (which, incidentally, is now illegal in many states while the car is in motion). So, charging for that felt a little slimy (even though automakers routinely charge for other safety features).
Furthermore, in addition to the safety aspect, with CarPlay, Apple saw the opportunity to get their foot in the door and claim ownership of some of the vehicle's screen real estate (not to mention location and other data). This is valuable position to be in strategically, and Apple may be planning to capitalize further.
Enter NG CarPlay which firmly has nothing to do with providing a distraction-free method of using your phono in the car. It is absolutely all about providing a more user friendly (or at least more familiar) interface to the car itself. And this feels very much like something that, were it eventually provided without the vehicle owner needing an iPhone, could be a good fit for a subscription service. After all, many automakers already use the subscription model for various in-car services and connected features. Except here, it'd now be an Apple-branded subscription, and yes, Apple would demand most of (perhaps even all of in some cases) the revenue from it.
So then, the answer to the question, "Why would Apple ever allow manufacturers to use its software as the default experience in their cars?" could be "Because it becomes a paid service". You choose not to pay the monthly fee, great - you get whatever native solution the automaker has cooked up (or thrown together as it can sometimes seem). Of course, the original CarPlay is still around and remains free if you have an iPhone. Meanwhile Apple's services are netting them billions of dollars per year, so they're clearly finding a lot of takers. And I suspect plenty would pay for this one too.