Just wanted to cycle back on this on more time: "Tab" should work normally, i.e. it should cycle through all focusable elements on the page. This is really important for accessibility; users who can't use pointing devices or touchscreens depend on this in order to be able to use a page at all.
Furthermore, there should be a mechanism readily available to turn these custom binds on or off, so they don't conflict with the user's own keybinds for navigating. For reference, Searx makes vim-like navigation opt-in. I think making these opt-in is a good move, since users are supposed to use Kagi signed-in with customizations applied.
I think some a11y conformance is an important milestone to reach before exiting beta, at least WCAG A (though sites ought to reach AA).