The satellite image you relied on appears to be outdated.  
You marked Way 4087817 with "access=private," but it is actually open to the public with no gates or barriers.  
Nodes 12117981770 and 12117981773 were added by you as gates on the ways, but these "gates" do not exist.  
It is possible to drive from Safi al-Din al-Hilli Street to al-Jihad Street through this area without any restrictions. In fact, I used this route several times just last week.  
Regards.  