> isn't h a consonant with its own distinct sound

Usually, including in “hypermedia”, but “hypermedia” is either an adjective or a mass noun, not a countable noun of which you can have a single instance.

“a hypermedia... ” looks like you are using it as an adjective to modify a countable noun, and when there is no noun looks like you forgot the noun; “JSON is not a hypermedia unto itself...” should probably be something “JSON isn’t hypermedia unto itself... ” (I’d actually prefer “JSON, on its own, isn’t hypermedia”, but that gets behind the issue with the use of the indefinite article.

It depends on the sound of the h. "A house" vs "an honor"

That's not the problem. It's similar to the effect of a garden path sentence: when I encounter the word "a" and then "hypermedia", I'm always expecting something to follow (e.g. "a hypermedia system", "a hypermedia format", etc). Encountering "hypermedia" as a singular noun makes me wince.

(The effect isn't ameliorated by modifiers, e.g. "a natural hypermedia".)