Ah, not sure I properly interpreted your original remarks. The original bridge_name tags (later, IMO incorrectly, synonymised to bridge:name) were definitely created to deal with the situation where the bridge has a name and the road has a name. People still often put the bridge name in the name tag on the highway, so I think retaining it on the way is important. 

If this name tag is moved to the man_made=bridge, then really it should just be name. :-)