Tag pages do require indexing, and your game didn't "disappear". It hasn't been reviewed by staff yet, because you published it three days ago. Please wait.
There is no "required" review process. If you saw your game on internal itch search, it was indexed.
But your game can be put on that review list after publishing for any number of reasons that you usually will not be told. You will not even be told that your game is on that review list, or that it passed inspection.
But there also is a possiblity that the recent changes in the search algorithm made your game visible for a short time, while being actually on that review list. There was something about the new search being asynchronous or something like that.
Either way, you still have to wait.