Right, but water not being readily burnt is not evidence that hydrogen is inert.

Yes. You are correct.

Thankfully kitchen sodium is in compound form, and thus not likely to react violently with water. In this context, the properties of pure metallic sodium are relevant because it would need to be handled in manufacturing. Kitchen salt is more commonly mined or extracted, requiring minimal to no handling of pure metallic sodium.

I hope this helps clarify any misunderstandings.

And both of them contain a bunch of protons, yet neither, nor their combination, behave like loose protons (aka Hydrogen).

But the Na has lost its electron to the Cl, so not the same thing.