No other way ?

[The Might]

The Misty Mountains are a different story, because of the snow and their height, they were harder to pass. And I think Gandalf wanted to go to Lorien before going further, so Caradhras was the shortest way except Moria perhaps, but we all know what Moria meant.

Now, it is quite clear that not all parts of the Ephel Duath were impassable. When the Host of the West refuses the terms named by the Mouth of Sauron we see Sauron's trap:

Down from the hills on either side of the Morannon poured Orcs innumerable.

Of course that are was easier to pass, since it was close to the valley, but still it shows that it was not impossible. I personally believe it might have been possible to pass the mountains somewhere near Durthang. Now the tower had initially been built there to make sure nothing evil comes into Mordor, however since it was close to the valley it couldn't have been a very steep area. Of course it would have been dangerous to pass there as well, but less then Cirith Ungol I'd say. Somewhere near Durthang they could have found some Orc-armour lying around...and the story goes like in the book from there on. That way there might have been no more army to besiege Minas Tirith...

And as far as the Morgai are concerned...it was not impossible. We see them coming to a ravine formed by the small water they encountered, so there clearly were ways of getting down. But they had other, more important reasons to keep going north. They knew they were being looked for after the incident at the gate with the watchers, so the only way was to keep going. And in some cases it would have been no use to attempt such a feat, since Orcs had raised their camps just below the slopes, and they would have noticed Frodo and Sam.