Surely the Fellowship took the wrong route?

[Inziladun]

The High Pass, through which Thorin and Co. had passed many years before, was surely considered by Elrond and Gandalf. 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 Gulder.
At the Council of Elrond, Legolas noted that:

'The dark things that were driven out in the year of the Dragon's fall have returned in greater numbers, and Mirkwood is again an evil place, save where our realm is maintained.'

The force of Elves in Mirkwood gave up hunting for Gollum because of Dol Guldur, and it would seem that having the Ring pass near it could have been quite dangerous.

Passing the Mountains further south at the Redhorn Gate should have been more secret, with the distance to Lórien and safety relatively negligible.