I guess it would be very difficult to create the internal structure, and especially keep the internal structure separated from the sidewall (which would make the entire thing undesirably rigid)?
Just add a valve that allows air out if the interior pressure rises more than x% above ambient atmospheric pressure, and allows air in through a filter if it's more than y% below. It wouldn't be that much more complex than the stem valves already added to tires.
The primary goal is to keep contaminants out of the interior tire structure, but a secondary marketing goal should probably be to make them look like normal tires at first glance. Wrapping the internal structure in something that looks like "normal tire" serves both.
Once you add in valves to let air in and out you run into the exact same fouling problems. Snow, small pebbles, road grit, any one of a number of things is at some point going to get sucked into that cavity. Normal stem valves avoid the problem by having a screw-on cap, which'd hardly work for a valve that needs to be passing air all the time.
