when I try to input the url for the search page in the launcher it redirects to the top games section
What would be the search page?
Because if you enter something like https://itch.io/search?q=halloween into the url box, that just works fine.
I do not know why the search box is not displayed on the Itch app's view of the page. And I also do not know why the search box on the left side is not doing what it says. It does not display the search results. If you enter a keyword, it will go directly to some game if you hit enter. Otherwise it will only display the top 1% of games or something like that, in a drop down box.
The app search bar also doesn’t show adult games
That's not true. Just browse the popular games. There will be adult games shown. Or type the beginning of the currently most top rated adult game in the search box of the app. e t e r is already enough to display the game in the suggestions.
Hmm. Curiously enough, that does currently not work on the web page. It appears with eternu in the results, but not as a suggestion even with full name.