The logical explanation is quite simple: thrones stay in play with durations because that's what the rules say

But in all seriousness, thrones don't move to other non-play areas when the card they throne does so I'm not sure why you would think they would go onto the tavern mat. For example, when you throne a card that trashes itself, like Embargo or Acting Troupe, the throne doesn't also move to the trash. When you throne a Horse, the throne doesn't go back to its pile. So why would the throne go to the tavern mat?
When you throne a ratcatcher, next turn you only get to call it once and trash one card, the part below the line doesn't get done twice, so there's no extra thing to remember about because you throned it last turn.
Also your example misunderstands how these cards work. When you throne a ratcatcher, you don't play it twice, then move it to the tavern mat, you play it the first time, putting it in play, and move it to the mat as part of resolving that play, then you play it again, but don't need to move it back into play because you don't do that for subsequent plays of the card via thrones (its just a rule), then while resolving the second play you try to move it to the tavern mat again but its already there.
Also, just to be clear, duration cards don't get played again next turn, they simply continue resolving the plays that happened the last turn, which haven't finished yet. The thrones stay in play to remind you the effect you are still resolving is happening multiple times.