My mind is really bad at reshuffle logic, but I have this theory that Inn's on-play effect doesn't really do anything if you shuffle in 5 cantrips into your empty deck and trigger the reshuffle anyway.
Like... if you didn't buy the Inn, those 5 cards would be in the next reshuffle, just randomly mixed in through the whole deck. When you shuffle those 5 cards in, you move them to the beginning of the reshuffle essentially, but they aren't part of two reshuffles, so it's like they're still getting their normal participation in the reshuffle. You're just eating your Halloween candy on your way back to the minivan.
Putting cards at the earlier part of the reshuffle could be useful if the game might end soon. But that's a condition that makes Duchy attractive too, and I've noticed at least most of the time when Inn is in the Supply, Duchy is too. If you're ahead, Inn is probably the lower variance option, if you're behind, it seems better to buy the Duchy and then pray to whatever you worship in order to obtain Inn's ongain effect in the reshuffle and save the 5$ that way.
When you shuffle in stuff that won't trigger a reshuffle, like three festivals and two monuments or whatever, you get two uses, which seems good. But even then, reshuffle logic is complicated and iterative and confusing, because you will reach the 2nd or 3rd iteration of your deck, when it is stronger, one hand slower, I think. I guess the clearest way to demonstrate this point is to say, what if you shuffle in four Squires and a Festival, you guarantee a 6$ hand, and you also get the Inn action card, that's cool. But what if, next reshuffle, there's a Gold in your deck that makes it possible you'll get your first King's Court, and what if the pivot point for either player this game is when their King's Court first collides with their Bridge? Maybe you don't want to be one hand behind on a shot at that, even if you'll get 2 Labs that aid the collision in exchange (the Inn, and whatever you buy with that 6$ hand).
Ok I'll try to stop hear before this gets moved to Articles, then gets removed from Articles for being amateur.