This is fixed now, with proper tags and an additional connector to the new cycle track.  So, cycle routing should work now.

There is a principle named "duck tagging" in OSM: if it quacks like a duck… then it is a duck.  So, despite it is legally a highway=living_street, tagging it as highway=pedestrian makes more sense, given that bollards prevent vehicles from using it altogether.  But whenever there are differences between official data and OSM, the best approach is to document it.  I filled the "note" tag on the ways to explain.