Here's another hint. A slightly less restrictive version of the set, which actually might have made for a better puzzle, would also include Quarry and Vineyard.
The amended list -
Beggar
Horse Traders
Lighthouse
Moat
Quarry
Secret Chamber
Squire
Vineyard
is still missing one and only one card.
The problem with puzzles like this is that while it may seem obvious to you that there's one and only one card that completes this set, chances are high that other people could come up with multiple cards that completed the set according to that same reasoning. Therefore, even if someone guesses the reasoning you are using, they won't think they've got the puzzle right because there will be multiple cards that could go along with this set.
University?
This might be the solution if the logic is "cards that have multiple functions, at least one of which is useless in a Kingdom with no action cards in the rest of the Kingdom." But Festival satisfies that logic too (and Vineyard's claim to satisfy that logic is tough, it could power up Silk Roads, for example).
If anyone posts "the common element of the current set is X, and therefore the unique card that completes the set is Y", either A) there's an argument that not all of the current cards have element X, or B) there's an argument that more than one card could be added to the set.