Hmm, Flag Bearer is a bit of a player-2 advantage card, isn't it? Assuming multiple do-nothing terminal silvers aren't something you really want in your deck.
One of my first thoughts was that all the artifacts confer a player-2 advantage.
But thinking about it further, my theory is that whether or not there is a player-2 advantage comes down to the opportunity cost of regaining the artifact (assuming you've had it some time before).
- Positive opportunity cost means player-2 advantage. At any time when both players have made the opportunity cost the same number of times, the artifact will be with whoever had it second, which is more likely to be player 2.
- Negative opportunity cost means no player-2 advantage. The artifact will go easily back and forth without either player needing to undermine their position to get it.
Flag Bearer, you have to gain or trash a terminal silver. That is usually a positive opportunity cost. There is always something better you could be doing with the gain (eg gain Silver).
Treasurer, you have to play a terminal gold and forego trashing/gaining a Treasure. That is sometimes a positive opportunity cost, at least in early game when you'd rather play terminal gold and trash a Copper.
Swashbuckler, you have to play a smithy while you have cards in your discard pile and at least 3 coffers. My hunch is that saving up 3 coffers is often a positive opportunity cost (you could have spent them to accelerate your deck some other way).
So my theory would lead me to believe that in terms of player-2 advantage: Flag Bearer - yes; Swashbuckler - probably; Treasurer - probably.
Of course, the magnitude of any such advantage is affected by the power of the artifact on a given board. On a board where it's not worth getting the Treasure Chest then the magnitude of Swashbuckler's player-2 advantage is 0.
[EDITED to recognise that using Treasurer to get the Key comes at the opportunity cost of trashing a treasure.]