I went to "game assets" and searched "Pokémon" and even tried it without the accent "Pokemon" but a number of results that ARE on itch.io are not being displayed.

This is one example: https://bellblitzking.itch.io/pokemon-sound-collection#download

This result does not even show up. Even when I tell it to search for "sounds". Only one result shows for "Pokemon" and "sounds" and it is not even a Pokémon related page.

There are other things that ARE on itch.io that do not show up in the website's search.

Not every itch game or page is indexed by itch.

You can read about the topic in this link:
https://itch.io/docs/creators/getting-indexed

Indeed, but that particular game asset pack is indexed. it also has a very generic name, and will be drowned out by others in a generic search. Please look up projects by category and tag instead if you don't know the exact name.