Uhm. It does work for site: itch.io
But if the engine would limit itself to itch.io/games, you would not search the game descriptions, only the stuff shown on browse pages.
It is as I feared. If you add /something after the domain, subdomains are ignored. You cannot find any game descriptions, if you add /games, since all games are on a subdomain. But also, since no games have a url starting with itch.io/games/ - you will only find browse pages with that. Which is kinda funny in its own way. Try searching site:itch.io/games hechelion.
The google help page did not make this clear, as they do not cover this particular example. They write http in front, excluding the subdomains from the other end as well.