[The Might]

You have misunderstood me Legate.
I am also asking if in general there wouldn't have been another way. Gandalf might have known as he was not too happy to hear Cirith Ungol when he met Faramir.
If we are to talk only about Frodo and Sam, I would think there was an even more important reason then their need for speed - Gollum. They now trusted him as far as leading them on the way was concerned, and they believed that Cirith Ungol was besides the Black Gate the only other way to enter Mordor.
And as Faramir and nobody else could prove Gollum wrong, they had no other way then to follow him.
But if Gandalf had remained with them, they might have taken another way I suspect.