I have an HTML game meant to be played in-browser. I have it set to "Restricted" but would like to have select people have access to it. It seems like the solution for that is to generate download keys.

However these do not work. Since the game is played in-browser, there is nothing to download,  the page simply displaying "Nothing is available for download yet."

If I set a password, this allows people to access the game from the main page, but I still would prefer to distribute it via download keys, as my use case is to have the game be exclusive to Patreon subscribers. That is, the game is restricted but all patrons should receive a key to access it. I know I could simply give the Patrons the password but it would be easier to just have them receive links that work with no extra steps. If this is not possible then I will go the password route but I wanted to do my due diligence here.

As the name suggests, keys only work if your game is not only playable on the web, but also downloadable. Basically, Itch allows a person to gain "ownership" only if it can be downloaded, and that's why you can't give keys if your game only runs via the web.

If you want your game to only be playable via the web, the only alternative would be the password system you mentioned (I personally didn't know about it).