Many of the paths that you have retagged in this set are not sidewalks, as defined by OSM.  In OSM, it is customary to distinguish between footways/paths and sidewalks based on their relationship to roads.  Sidewalks, as the name implies, run alongside roadways, so the footway=sidewalk tag is not really appropriate for footways that cut through college campuses, parks, etc.  

For further guidance: 

https://wiki.openstreetmap.org/wiki/Key:footway
https://wiki.openstreetmap.org/wiki/Sidewalks

Also, you can use the OpenStreet Browser (https://openstreetbrowser.org/) to confirm that other mappers (not just me) follow this practice.  You can view the OSM footway data by selecting Transportation > Walking > Footways.  In well-mapped cities, like San Francisco, London, Berlin, and Toronto, there's a fairly consistent tagging distinction between sidewalks alongside roads and footways elsewhere.  