For people who want to have a 'solution' before the fix gets rolled out:
You can add the following to your custom css in the kagi settings.
#icon_new_ia {
display: none;
}
This will hide this new icon, but you can still click on the place where the icon was and see the dropdown, this way it does not destroy the search page design and you can still use the things from the dropdown if you want to.