I'm a little confused about what happens to the Mist Horror after you defeat it. The card text says, "After you act, shuffle this card into a random open location, then move to a random location."
At first I was afraid that this meant the Mist Horror wasn't banished, but thinking about it this doesn't seem like it would be the case. It seems like the Mist Horror should be shuffled into a random open location after you act, but then the encounter resolution would happen as normal so you would have to look through the location to find and banish the Mist Horror. There's no would on the card, so it doesn't override normal resolution.
Can I get some input on this card? Because it doesn't seem like that would be the intention.
Is this an application of the impossible rule, where shuffling it into the deck makes it impossible to banish as normal so you don't?
Yes. You can't banish a card you can't see, and you can't see cards that are shuffled into decks.
Skizzerz is right you never directly banish the Mist Horror.
Usually you get rid of it because since you keep shuffling it in locations at some point it will end under the henchman of a location, and if you defeat the hench and close, byebye Mist.