[JennyHallu]

Had they taken the High Pass, they would have come out on the other side of the Mountains far to the north. While the Pass itself might not have been closed, to travel south from there would have required passing dangerously close to Dol Guldur, an obstacle of the caliber of Minas Morgul. The Redhorn Gate left them in the uninhabited, neutral lands between the mountains and Lothlorien, all together a route less likely to draw Sauron's Eye. And as for the choice of Moria, Gandalf had passed through before safely. Were it not for Pippin and the well, it is likely that they could have gotten through in total secrecy, by an unguarded way.