Am a writer. To illustrate my point by analogy, I need a name of a famous mountain where experienced climbers know of only one route to reach the summit -- where climbers would be foolish to try any other approach than the one tried and true.

I can't think of any. Even mountains that have only one climbed route still have other potential routes.

With the right technical gear most peaks have multiple routes. But if you exclude ropes & pitons - the route with fixed cables on Half Dome in Yosemite is pretty much the only way up. Unless you're Alex Honnold.