I think with any install of BitWarden, be it a browser add-in or separate app, the one you are adding a new credential into knows enough to sync to the cloud, but the others won’t know that new data awaits in the cloud until they do a scheduled query/poll or you manually sync through those clients.

Having a push feature only works if you can engineer your app or add-in to open up the necessary ports or tunnels in the OS itself. Polling on the client end will always be easier to implement.

Also: just checked BitWarden v2.8.0 (449) on iOS 14.4, no setting for “swipe to refresh” anywhere in its settings.