Why Moria?

[Formendacil]

Originally Posted by Tuor of Gondolin

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).

I fear that the Sea of Rhûn route would have been a poor idea for the same reasons as the Drûwaith Iaur route: too long. The Fellowship had to get to Mordor ere the West was lost.

Also, the Beornings may have held the upper Vale of Anduin, but to get at Mordor's "back door", the Fellowship would have had to pass through or around Mirkwood (which in addition to being the home of Dol Guldur was also a warzone). Then they would have had to strike out across Rhovannion, which was probably swarming with Easterlings and Orks, and then into Rhûn proper, which was not only full of Easterlings, but a part of Sauron's empire.

The route that Frodo ended up taking may have been dangerous, but I think an alternative route in Mordor- which would have been too long anyway- would have been far more perillous.