[burnished]

First, this is an exotic material, so you should expect some weird stuff.

Second but probably more importantly you are correct that protons and neutrons are strongly bonded as part of a nucleus, what I think you are missing is that hydrogen molecules are essentially free-ranging protons and do not experience nuclear bonds with oxygen molecules as part of water, they are instead electrically bonded. It is not incorrect and even somewhat common to refer to hydrogen as being a proton, because typically it is. Sometimes you'll get heavier hydrogen with neutrons, but mostly you don't. Hydrogen also has a difficult time trapping and keeping electrons, so seeing it as a sort of 'hanger on' proton when its bonded is normal too.

I'm not aware of any thing that we humans use where a proton is the primary charge-carrier. I also won't speculate as to what sort of material properties they would have. If your thesis is 'protons as charge carriers is extremely unusual' then yes you would be correct. The reason that 'most substances conduct electrons' is because metals already conduct charge, which you are probably familiar with as the 'sea of electrons' theory, and semiconductors also use electrons or electron analogues as their charge carrier.

Does that answer your questions/doubts?

[perihelions]

Proton conductor:

https://en.wikipedia.org/wiki/Nafion

[burnished]

Oh super neat, thanks! If I understood correctly its more like a cation conductor than a specifically proton conductor, but much of the chemistry was beyond me.