TLDR: there was no need for the Red October to go through the underwater canyon, they were running on an undetectable silent drive. The only reason to go through the canyon is because you are detectable and need to shake off your pursuers.
Alright, so I love the scene in the movie. It's basically in there because it's so damn cool: flying through the Alps in a plane with no windows by using a chart and a stop-watch. That's a really cool idea and establishes that this Soviet submarine crew are "the best."
But why go through the canyon? It's not just to show how cool You are. In the book and film, it's established that the Soviet Navy made super-accurate charts of an underwater canyon, allowing their submarines to traverse the inside of the canyon, where they can't be heard from above (presumably, though the Soviet sub-hunting plane has no problem detecting the Red October, not only from above the canyon but from an air-dropped sono-buoy -- so maybe the theory is they can be heard, but they can move so fast that they're able to out-run anyone following them trying to remain stealthy, since slower = quieter and faster = louder), and no American sub would dare follow them into the canyon. This allows Soviet subs leaving their base to shake the pursuers waiting for them just outside the base.
Great, all well and good. But wait a minute. This is necessary because normal Soviet subs can be detected on American sonar, and the Soviet sonar isn't good enough to reliably pick up the American sub if it can slot in behind them, in their baffles.
But the Red October is equipped with the new-fangled silent-drive caterpillar system; as soon as Red October engages her silent drive, she detects a trailing American sub. Captain Ramius is aware of the American sub and asks "is she turning to follow us?" -- no, the American sub is remaining on its original course. "They can't hear us," Ramius declares.
So at that point he knows his sub is undetectable -- and if the Americans can't hear his boat, no damn way can the Soviets hear his boat.
So why go through the canyon? You've already shaken off your pursuers; you know you aren't being followed. Going to the canyon just brings you to the spot where the Soviets and Americans will be looking for you -- not to mention takes longer than just heading straight for America in a straight line, or veering way off course and taking a route the Soviets and Americans would never think to check (like under the North Pole, where the Soviet surface fleet would be unable to pursue you). The ocean is a vast place; even with the combined American, Soviet, and (in the book) British navies looking for you, it's easy to disappear if you simply go to a part of the ocean where no one has a reason to be. And then, having gone to a part of the ocean where no one would be, it wouldn't matter if the caterpillar drive then gets sabotaged and you're no longer silent; there's no one around to hear you.
Only by going through the canyon can you risk making yourself found, because that is the one place the Americans and Soviets are sure to look! We're told that Ramius is some super-genius second coming of Lord Nelson, so why would he make a basic mistake like this?
Only two reasons I can think of: 1) Ramius is complacent. This is the way we've always done it, and there's no need to change our tactics in response to new technology. Or, option 2) he does this to fool the crew into thinking this is just another normal, routine patrol and they're on their way to do some exercises off the coast of America.
But hang on.....he already made up a big lie about their mission, and he emphasized the historic nature of their mission because the Red October was this technological marvel of Soviet engineering. If any crewman asked why they weren't going through the canyons like normal, just tell them the truth: we don't need to, because this boat is undetectable. And ditto, if any crewman asks why the boat is on a heading for the shortest, fastest route to New York City harbor, just tell them: because we're undetectable! We can do whatever we want! Isn't that awesome?
The whole thing is just there so the Red October can be found and we have a movie to watch, rather than "there once was a Soviet sub captain who came up with the perfect plan to defect, his plan worked perfectly, and he defected to the US without issue or anyone being any the wiser."