Hi I am a Vivaldi employee so perhaps I can answer this one. This is not a bug in Kagi's extension, nor a bug in Vivaldi for that matter, but rather intended behaviour. The issue here is that Vivaldi does not allow extensions to set the default search engine without one further tweak. We did this intentionally because some extensions (not nessecarily Kagi's—which is quite clear in its intended use) abuse the ability to redefine the default search, without adequently informing the user that they have done so. So it is possible but we do require you to take one extra step.
Look again at the screenshot you provided and notice this option "Settings → Search → Search Options → Search in the Address Field→ Allow Control by Extensions". Tick this and restart the browser and then you will get the desired affect.
P.S. Adding Kagi or any search engine by hand is perhaps easiest as you seem to have found. For others reading this all you need to do is right click on the Kagi search field and choose "Add Search Engine…"
Then in the popup dialog check the option to set it as default.
As you noticed there are more comprehensive options under "Settings → Search".