Pfew, that's quite something that you've cooked up!
The search-as-you type is definitely nice. However, I'm not quite comfortable with the overall setup.

I feel like the things like toggling the result types, adjusting the theme (colors and light/dark mode) and mobile support are things that go beyond "I'm doing a little tweaking" and should rather be proposed to upstream. Maintaining this exclusively for/as part of PTB is not something that I would look forward to.
Recall that we switched from the rtd theme to furo exactly b/c we were starting to fiddle with small details in the theme :D

Yes, I did think about writing an issue for this but then I was not sure what I would ask them since:

1. The official RTD theme doesn't actually support dark mode, let alone them modifying their design for e.g. Furo's built in dark mode. Switching on darkreader just fixes it for both cases
2. I could ask for advanced search, but that still wouldn't support parameters, since that relies on sphinx-paramlinks.
3. the upstream version does support mobile properly, it's just my modding has changed the position of the elements (you have to scroll right to access the dropdown menu). I'll try to fix this, should be simple

So I decided to just fork and make own changes. Maintaining shouldn't be too much of a problem? Worst case would be that it would just fallback to regular default search. I do think here that the positives outweigh the negatives

I see your points, but have a different view point regarding the negatives & the positives … at least until support for mobile devices is enhanced (apart from the "search for" toggle being way off to the right, the search box is "striked through" by the blue separator bar, at least for me)

LGTM :) @Poolitzer do you want to have a look as well?

Just from a quick look at it, the type button looks misaligned. And it doesn't say no results if I filter out existing results for my query.

Yes I like the CSS look now good job! I just realized going backwards/forwards one page does not show the search again, is that fixable? Even worse if you filter stuff out, all of that apparently generates a new page. Makes you think the browser broke :D

Almost happy. Back and forth works great now, but if you

1. type something in the search bar
2. press enter so the "old" search site is opened
3. now press the search bar on the left again

The search is not triggered. Is that fixable? Then I am happy.