I doubt it would end up being lighter in an aircraft when you add in the required liquid cooling system that would go with it. Plus if you have a decrease in efficiency (which I would expect), there goes your weight benefit.
The heavy fuel is interesting. Fundamentally, there's no reason why that engine should run kerosene/Jet-A/diesel any differently than a conventional piston engine. In other words, spark ignition it won't work well, and compression ignition you'll get back to the same weight issues with diesels. So I don't see that being a game changer. It's interesting that the one application I could find for a swashplate engine was in a torpedo. One time use, doesn't need to last very long, highly disposable.
All of these odd engine designs have existed in various forms sometime in the past century. None of them have been compelling enough to cause a major change in the commonly seen internal combustion engine design. That should tell you something.
Still, it's cool to see people trying. I would rather see the FAA let me write "EXPERIMENTAL" on the side of the 414 and let me play around with the TSIO-520s. I could net significant improvements...