I use Kagi to search on my own static blog (www.ploum.net ) through a lens.
I find it so cool that I would like to offer it to the visitor of my website. They would click on the "search" link on my page and Kagi would open with a "forced lens", allowing them to search through my domain.
Benefit includes increased visibility for Kagi and one more valuable feature for paying users.
Technically, this would be a lens available also to not-logged in users. We can imagine a fancy url like : kagi.com/public/ploum.net/
In its "lens" list, an user could get an option to make it public trough a customizable URL.
To avoid abuse, one could imagine only allowing "verified-domains" in public lenses, where "verified-domains" are domain for which the user has proved ownership (through uploading a file or through a DNS record).