first of all, on OSM we don't map for data consumers, if an app is using the OSM data incorrectly its on them to fix their implementation. I have mapped enough pedestrian and foot paths in Miami that look the exact same to be 99% certain that this is a pedestrian and foot path along with driving by many a time as I used to live somewhat near by. again designated does not necessarily mean public access, it just means that the specific type of traffic (in this case bicycle and foot) are allowed to use the path, again the access=private is what marks it as private.
again as previously explained desginated only tells you which transportation modes are allowed to use a specific way, private, permit, etc. tell you what the permissions on the way are, aka. if you need permission to access a way. 
again access=private already tells you that, while changing bicycle=designated to bicycle=private actively lowers the amount of information encoded as we are no longer specifying that this way is specifically for pedestrians and bikes. again let me make it abundantly clear that these ways are tagged as private due to the access=private tag. 
also don't pretend I don't know what the underline is as I have made several fixes to its alignment after the second phase of construction finished up last year.