Still searching ... Rated 1 out of 5 stars
You Write "... for a specific usage and the addon does not fit your wishes ..."
My idea is: Not so VERY specific (a search-url in a new tab) and I dare guess the wish of most people (focus on the search type-in line).
Now: I have to grab for the mouse to click in the type-in line. (or tab-key umpteen times) when the add-on just saved me a grab for the mouse to click on home-page icon.
Do you know of an about:config line that governs focus?