Yes, that's always been my question also. Gandalf's response to the news that they were going that way - "why that way" suggests he has some other idea in mind. But black gate is obviously impossible. And journeying all the way round was obviously too far in their state. What other way could there be which he would rather that they had taken?
I don't think the book itself suggests any other way. So, if there was another way we have to suppose it wasn't written into the book.
As an "In universe" expalnation - since the book represents the story as written down after the events by the Hobbits themselves, that could just be because their geographical knowledge of the range is incomplete, and nobody thought to ask him about other ways - or never wrote it down.
As for whether a mountain range could be so impenetrable as that in real life, I'm not sure. I think that for instance some ridges in the Alps can't be crossed on foot without a rock climb or scramble likely to be beyond the abilities of a hobbit, especially one afraid of heights like Sam (except for Gollum).
But am not enough of a mountaineer to give examples. In the Himalayas for sure - but that's a bit different because they are so high they have snow and ice and strong winds and basically impassable for anyone except those with good equipment except for the lower passes. The hills around Mordor are not as high as that.