ROB,
But they also wear a tails when not in a shul.
The point is that the Tallis is not because of the shul, otherwise everyone would have to wear a talis. And you would have to wear a talis anytime, not just by davening. (Although Avi Weiss is notorious for wearing a talis at political demonstrations).
The reason has more to do with kavod hatzibur. And in Weiss’ case, the Tzibur was largely a baptist group so kavod hatzibur with a talis or yarmulka is moot.
Anyway, you are so missing the forest for the tree.