Surely the Fellowship took the wrong route?

[Zigûr]

Originally Posted by Inziladun

I think it was rejected because of the likelihood of Orcs being present there, as well as the idea that if the Fellowship used that pass, they would have been obliged to journey south under the watchfulness of Dol Guldur.

I think this is it. The "culling" as it were of Orcs in the Battle of Five Armies had been short lived, and I think the danger posed by Dol Guldur cannot be stressed enough. That being said, the Beornings apparently kept the High Pass clear, but that leads me onto another matter:

Originally Posted by The Mouth of Sauron

They could also have had the help of the Beornings and the Woodmen of Mirkwood

Personally I don't find this idea especially likely. The Beornings and Woodmen were trusty and hardy folk in their way I'm sure, but I don't see them having much to offer the Fellowship. As we know from the report of Glóin the Beornings charged tolls to pass through their land, so it seems to me that there's more evidence of them being a hindrance (however minor) than a help.