Go ahead and look up a Bib Mousse.
A closed cell foam can be altered in many different ways to adjust the performance characteristics of the tire. Consistently and inexpensively. That is why we use it.
The roar rule about not using air pressure is likely a cost control measure from off-road. When we transitioned from hard rubber tires and no inserts to soft rubber and foam inserts there was a time when it was a case of the haves (factory drivers with soft tires) and the have nots (anyone without access to the soft tires). Eventually everyone transitioned to the soft tires, and to prevent air pressure from possibly becoming the next step in the evolution (and a MASSIVE degree of added complexity and cost), roar made the rule.
You have to remember technology has come a long way since then, so the cost on some of the equipment required to set the pressure accurately may have come down some. There are manometers that are $150 dollars, and there are ones that are $2000+ Add in the development cost of tiny air valves that could be reliably implemented onto current cars would add more cost too.
Could it be done as a technical exercise? Maybe. Would it offer a gain worth the cost and compexity? Not likely.