Surely the Fellowship took the wrong route?

[The Mouth of Sauron]

Setting off from Rivendell southwards was considered more secret and safer than other routes, but surely this was mistaken. If the Company had crossed into Wilderland east of Rivendell, they could have avoided the Redhorn Pass, Moria, Isengard and the Gap of Rohan.

They could also have had the help of the Beornings and the Woodmen of Mirkwood, and come to Lorien by foot west of the Anduin, or even by boat from say the Carrock, which was a shorter river journey than the one they eventually took from Lorien to Rauros.

[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.

[Zigûr]

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:
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.

[IxnaY AintsaY]

Perhaps the High Pass is called "High" relative to the other option. I wonder if it may have been at a higher atltitude, and thus, maybe, even more perilous to cross than the Redhorn in the depths of winter.

We also no little (I think) about the land immediately east and south of the High Pass and west of the Anduin. Perhaps it is especially rugged, and it's certainly easy to imagine the marshy areas around the Gladden aren't a king's highway. Perhaps the region is also thick with Sauronic forces amassing for their eventual attacks on Lorien.

The land between Mirkwood and Anduin does seem passable, based mainly on Isildur's path after the Last Alliance, but obviously not without passing very close to Dol Guldur.

[Farael]

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!