It wasn't "easily" detectable. Anything that moves has a sonic signature, but the Caterpillar Drive would have been very difficult to detect compared to a standard prop driven sub. Remember that the sonic signature was almost unreadable until a large segment of it was recorded, then played back at 10X speed (which suggested a man made pattern of sound), something that can't be done in real time.

If you know it's there, you can bang it with active sonar which will identify not only the sub but also the sub that's sending the sonar.

Even stealth aircraft can be detected, IF one knows the general location of the craft to start with. The Dallas had trouble tracking the Red October even though it had a good idea where it was.

First off, I want to get it out of the way that I like this movie and have watched it many times. That said, the movie does sabotage itself numerous times. First off, within minutes of Ryan presenting the spy photos of the Red October to Tyler, they figure out exactly what the strange doors are - no research or consulting with perhaps a guy they have on the inside of sub development in Russia or anything, Tyler just pulls it out of thin air "Hey, this is a Caterpillar drive." Second, the Dallas is not only in the right place at the right time when Red October seemingly up and disappears, the guy manning the sonar is some kind of savant who can see past the Red October's trick and is, as fatdogtavern points out, easily able to detect the undetectable - it'll slip past early warning and SOSUS, remain completely undetectable parking a couple hundred warheads off New York ... unless your name happens to be Seaman Jones, then you're screwed - this guy can be standing in Time's Square on new years eve and detect a mouse farting in Australia. Third, and this is more of a story telling choice rather than the story itself, the movie sets it up that oh no, is Ramius a mad man who wants to use his bombs or does he want to defect? Only to answer it 2 seconds later with the scene in the Red October's Mess where it is revealed that they do want to defect, completely defusing any tension the movie had built up.

The last point I can look past but the first two have always bothered me. Oh no, this scary new Russian sub is so big and what are those mysterious doors? I'm going to take a completely random and wild guess based on zero evidence and say they enclose some strange, undetectable kind of propulsion! Oh wait, not so undetectable after all, even before we've had a meeting about it, one of our guys has figured out how to track it. As I said, I do enjoy the movie, but there are just too many eye rolling "how convenient" moments for my taste.

You're really overthinking this. Back in the old days, there may have been several scenes of exposition, but today that will drive millennials nuts.

I never read the book, but IIRC, Tyler said the U.S. had been working on, then abandoned such a propulsion system, so he would have been aware of the huge doors that were not torpedo tubes needed to move the vast quantities of water through the hull that a caterpillar drive would require. What else could the doors be for? Ryan guessed missile launchers but it was explained that horizontal launching wouldn't work. Could they have added more exposition? Sure. Would it improve the pace of the film? I don't think so. If the U.S. did have spies placed that tipped off about the caterpillar, wouldn't that have really deflated the story?

The RO was still nearly impossible to track, even with Jones' brilliance. If the Dallas didn't know where the RO was, they couldn't find it unless they pretty much tripped over it, which is what they did. If Ramius suspected he was being followed, he could have simply changed course and the Dallas wouldn't have found him even with Jones' skill.

It wasn't until the caterpillar was sabotaged that the RO was trackable by both sides.