Eh, I have sympathize with both. Bridges are a pain in the ass, they are choke points for traffic. As a cyclist it can be more than a slight disruption to your flow to have to stop and get off and walk your bike through a crowd.

1) Anytime you come to a complete stop your body temp rapidly rises, and your body starts shutting down from the exercise. Making it physically taxing to get on and off and on again.
2) You are now a bigger target, before you where one person width going slightly faster than pedestrian traffic (you slow down because you don't really want to hit people).
3) It is more cumbersome to push a bike than to ride it, which can sometimes disrupt the flow of traffic more than if you just ride it.

At the end of the day, its time they improve the bridge and make more room for both pedestrians and cyclists. Which from what I understand NYDOT is hoping to do https://nyc.streetsblog.org/2017/12/08/dot-might-widen-brooklyn-bridge-promenade-but-rules-out-claiming-a-car-lane-for-bikes/