I'm curious if you have any other purposes for the "Birthright mat" - is this part of a set of cards you have? Otherwise, I'm assuming the only reason it's "their" mat is in case they set aside an Action-Victory card instead of just an Action card (or else it could be a communal mat). Alternatively, couldn't you just say "Put a non-Duration Action card from the Supply on this"?
To be honest, I did not even consider a communal mat. I like to start with mechanics already in the game, and I am not aware of such a communal mat. The reason I did not put it on top of or under Birthright itself is that I wanted to borrow the mechanic from Inheritance of the card staying out of play (so that a player could use a card like Experiment). Inheritance uses a token, but since I wanted to have the possibility of multiple cards, a mat seemed like a pre-existing solution.
Would the card allow you to use Command cards, or should they be excluded like Durations?
My first thought was that it shouldn't, but when I thought about it I changed my mind. My understanding is that the reason Command cards don't play other command cards is that they could, in theory, create loops. This card does not play the selected card on that turn, so even if it uses a Command card, it couldn't create such a loop (as far as I understand it). If there is another reason not to use a Command card, I would appreciate the input.
My initial thinking was that the fact that the other players got to use the card as well means that most of the restrictions of the other cards that play other cards (Command cards, Inheritance, Prince). That might not be correct, and I will have to think more about it.
This card could really slow down the end of the game if each player starts getting multiple cards set aside. Each start-of-turn would be playing a good handful of cards before they even start their Action phase. To keep it a little more subdued, would it work better as a project (on purchase, when you place your cube you would also choose the card to set aside for it)? Or a "once per game" event?
This is something I had not thought of. My inclination is that this is unlikely, as playing this card effectively is hard enough that I don't think multiple players will do it multiple times. I also don't think there will be much incentive to play Action cards with a lot of choices, as that will make it much more likely your opponent can use the card effectively. But I will definitely consider this part and think about reworking it.