I think the following tags belong (mainly) to the crossing node, not to the crossing way:
crossing=unmarked
crossing:markings=no 

I've kept them on the footway (but I think they don't belong there) and copied them also to the crossing node. And I've added the highway=crossing tag to the crossing node which was missing.

And since the south end of this road is still unconnected to the footway I've added a connection. Hope this is correct.

Thanks for your effort.

Regards
Alex