Why Moria?

[Tuor of Gondolin]

Why would they risk harsher weather by going further north where it was more common, and take the longer route in face of impending doom?

To interject on this point, the northern route (east to River
Running) while longer and perhaps minimally colder (Caradhras
was a tall peak so even its regular winter passages might well
be as difficult as the High Pass). Also, the route east
was generally secured by the Beornings, plus they could approach
Mordor from a presumably unexpected point (The Sea of Rhun)
as opposed to knocking on the front door (rather unwise).