I agree the lead at top of verge could be dubious, it must be getting in on that dry verge somewhere. Can we have a PIC of the return lead up the top there. I can see all the fixing attempts they tried! Wow.
Osb3 followed by EDPM could be the way to go, worked well on a job I did last year. The OSB would probably need to be 4 sided tongue and groove. Screwing it down to the concrete joists doesn't look fun though. You could fix 2x4 to the side of the concrete and screw to that.