Not sure what the problem is in those pictures. Looks standard to me
Sure the vapor barrier sounds weird, but you've got:
-Insulated metal exhaust duct (check)
-Hole in roof (check)
-Bird Screen (check)
Most roof vents are literally just holes in the roof with the angled cap on top. No flaps no nothing, just a screen. The damper to keep cold air out is generally at the fan itself. The exhaust vent just gets directed out that hole, but by no means is it usually any sort of "sealed" connection to the roof vent. Don't see any moisture issues on the plywood so roof connection looks A+.
My only guess would be either snow getting blown in, or the fact it is such a long vertical run that a decent amount of moisture is condensing/freezing on the sides and then melts when it warms up (weather wise or the next long hot shower).
Could try further insulating the ducts or adding a sort of drip catch (ie a longer horizontal run, or something almost like a drain P-trap).
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