It looks like roads here where tagged with "vehicle=destination" (e.g. https://www.openstreetmap.org/way/851256959) approximately at the same time where you created the note.  To make it correct, "bicycle=yes" and "horse=yes" would be desired.

So perhaps that the best option would be to use those tags for the entire local grid here, and ignore the fact that city authorities will occasionally fail to put the additional sign on one of the roads.

By the way, this is an interesting example where we can already see the new rendering style of "vehicle=destination" with dotted lines on osm-carto.  (Until recently, only "access=destination" was rendered.)