I don't have any images handy, but there are fan designs of the Constitution bridge that have the captain and helm facing forward, the main elevator shaft on the center line, and the elevator doors offset to port. So, an elevator would make a slight jig to port when going to the bridge. One design I saw even has an extra car on standby stashed in that space.
The Starship Exeter fan film "The Tressaurian Intersection" has the motion indicator on the elevator showing this a slight horizontal jig when it reaches the bridge, indicating this centered alignment.
The TAS bridge has a second elevator on the port side as well, without any external indication of this difference with the TOS bridge, so there's no reason why its bridge can't have this centered alignment.