Could you consider simply removing the "display: none" on .index_page .index_sidebar .game_search in the stylesheet as a short term workaround? It does seem that search works in the box that appears when you do this.
I am sure that this is costing itch users - I'd have moved over to another site if I didn't know I could get the game I want on the website.
The search box on the app doesn't work, as other people described in the thread (namely it fails to actually search). In my case I typed "spacechem" into the only visible search box in the app. I simply cannot get it to perform the search - pressing enter shows me a seemingly random game. Other people have posted screenshots of the issue in this thread.
All I noted was that I could perform a search in the app by reenabling the search box. The "index_page .index_sidebar .game_search" is a CSS selector that refers to the search box that has been hidden from the sidebar in the app.
That said, it appears this has been repeatedly reported as broken, eg https://github.com/itchio/itch/issues/3095
