Surely the Fellowship took the wrong route?

[Farael]

Originally Posted by The Mouth of Sauron

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

I think this is exactly the issue at hand. We know that the Fellowship gathered in Rivendel and they may have intended to rest in Lothlorien, but even then I think it's stated that Gandalf hadn't discussed his plan beyond crossing the Redhorn.

I think that Gandalf intended to keep his interactions with other peoples to a minimum. After all, they may have been able to ask for help from (say) the Beornings... but could they really tell them what they needed help with? Could they trust them with the knowledge of the ring?

All routes had their dangers, but they were hoping to take this (somewhat unexpected) path to avoid notice by foe... and friend. And let's remember that Gandalf was somewhat eager to go through Moria in order to escape any trackers, so he was further betting on going "off the grid" rather than asking for help!