I've tried with BRouter: https://brouter.de/brouter-web/#map=19/51.48301/11.96895/osm-mapnik-german_style,terrarium-hillshading&lonlats=11.968687,51.483042;11.968368,51.483112&profile=hiking-mountain

Indeed, it does not take the obvious short path. The tags seem correct but I think the path itself is not correctly connected to the "Fußgängerzone". I'll investigate...