Vlad I have a suggestion: an option to differentiate between searches made from the browser location bar, or from the Kagi search page.
I was thinking about the fact the before Google launched Chrome, we had two text entry fields on the browser shortcut bar: a location bar and a search field. Google started the trend to merge these (obviously to their own benefit, as it made Google the default way of navigating the web).
If we think about breaking out these two functions again: the location bar is for finding and visiting a specific website, the Kagi search box is for text searching/research.
So the differentiation is not "searching big sites" vs "searching all the other sites", it's "getting to one specific site that I don't know the URL of" vs "searching web content".
I feel I often waste a Kagi search when I know what site I want but I've forgotten or don't know the URL. In these cases I don't need more than the most basic and shallow index of the website content: probably just the headers and footer. Often just the URL without any content at all would have been sufficient.
So Kagi would need a broad but extremely shallow internal index of the whole web, as opposed to a narrow but deep index of the major sites. (This whole suggestion hinges on the idea that such a shallow index would be feasible for Kagi to maintain, I have no idea whether this is true...)
A couple of real-life examples: I haven't looked at Kagi feedback for a few months, and I couldn't remember the name of the website. Was it kagisupport, kagifeatures? It wasn't coming up in my history drop-down in the location bar, so I did a search for it.
After I've posted this, I need to visit my local council website. It's called something like "inner west council". But I don't know the exact URL, so I have to search for it. (I just searched on Kagi, it's www.innerwest.nsw.gov.au).
In your example, someone typing "twitter" in the location bar wants to go to twitter.com, but if they type "twitter" in the Kagi search field they want to research news about twitter.
So my suggestion is: in the Kagi search plugins, have a user-selectable option to make location bar searches go to the "basic results" page, which has Kagi's top 10 guesses for the site the user was looking for, and a big "continue your search on Kagi" button. These results are free.
If the user wants to make a regular Kagi search, they can do it in one of three ways: launch the search from the location bar and then click on the "continue" button; navigate to kagi.com and start the search there; use a bang.
Users who are on metered plans and are price sensitive, like me, can turn on that feature. Everyone else can ignore it. You could pro-actively promote it to users who use up their quota.