I rarely use bangs and if I do it's only bangs I learned from others (like the Discord server or the help docs). I really want to use them and be more efficient, so I propose adding the bangs as hints to search results.
A user searching for something (like in the examples given "kagi inc") would stumble on results and see how they could have searched on that site itself directly. That way the user doesn't need to learn the bangs by opening documentation, but just see it and eventually remember and start using it.
Not everyone will like this, or people advanced enough don't need it any more, so it should be toggle-able, preferably without the need for custom CSS.
Below is a little mockup I made by inspecting the web page and adding the bangs manually. I'd consider a popup that explains bangs briefly and lists more bangs available for the domain and subdomain of the result, if applicable.
Here's a before too:
