The suppress from homepage doesn’t remove the category’s posts from appearing on the Latest page. That would certainly help, or if there was an alternate way to do it, or create a page similar to “latest” that could exclude certain categories.
I’ll work on a sample of the auto-suggest for mobile. I know Discourse auto-suggest on the desktop search, but the mobile interface doesn’t really allow for it. However, if the auto-suggest on mobile just included categories, that would help communities with larger number of categories.