I disagree. I only clear login cookies created in the last hour. Since Kagi login happened longer than 1 hour ago, it is expected to remain valid and functional.
But it seems that Kagi somehow recreates its login cookies at every search, which seems unnecessary and does not work well with Safari time-based cookie pruning.
Kagi is the only website I know that gets its session invalidated by “Clear 1 Hour” action.