Loading a page to search I aim to paste/type immediately, which needs the cursor in the search box.
works:
B, DDG, Ecosia, G, Neeva, SS, You, etc.
web.archive.org
WorldCat
Github.com doesn’t paste, and doesn’t show a blinking cursor, but it does put typed characters (after using the ‘s’ shortcut**) into the search box.
Stack Overflow doesn’t have the cursor in the box, and also forces (those not logged in) into a CAPTCHA flow.***
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* These each do use OpenSearch Query Syntax(?) that enables search from the address bar.
** I stumbled on this, tapping frustrated, then confirmed.
*** You must “Enable keyboard shortcuts” with a profile via settings to enable ‘s’ to focus on the search box in Stack Overflow.