You're right. I tried it on osm.org, and they do all route across it.
I was using OsmAnd~, an Android app from downloaded F-droid. It will route across it if it's a short enough distance, but if you tell it to route walking directions from the sidewalk in front of Lexington to the sidewalk in front of the Village, it will have you cross the street twice to avoid that section because there isn't a crosswalk annotated there.