This seems to be impossible, at the moment. (To use specific keywords: You're asking for a way to make any searches (not just through the InputBox extension) send you to the "Content" tab (&profile=default) instead of the "Advanced" tab (&profile=advanced)
The tab you end up on, is based on whether we individually (as logged-in users), have ever selected the checkbox "Remember selection for future searches" in the "Advanced" search tab. -- You can confirm this by doing a test search in a Private/Incognito window.
I've filed phab:T170011 with the request for a method to reset it. (I believe it might still be possible by resetting all user-preferences. But that's obviously not optimal.)
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Semi-related: There's ongoing work at m:WMDE Technical Wishes/AdvancedSearch to rethink the Advanced Search - both making it clearer, and making more of the advanced functionality easier to use. I'll send a note to one of the people working on it, with your points, and ask if there are separate mockups for the mobile UI, yet. Thanks!