alt Hacker NewsI really doubt it, and I am not surprised that apps didn't work. I also don't think playing a musical instrument actually gives as much insight here as you want - it's a very indirect transfer if you're learning via classical methods or if you are learning purely by reading. Jazz and contemporary methods involve a lot more "listen, then play back" rather than reading. If you want to work on ear training and theory, cross-training with some jazz or contemporary helps a lot.
The way music schools teach this is relatively brutal and annoying, with a _lot_ of repetition and testing (eg "sing a major second above this note" and "identify the interval" questions), but I am not sure any other method works. At the same time, everyone going through an ear training curriculum does pick up decent relative pitch. This can take a year or two for college music majors, so it's not exactly a casual exercise. However, I assume the major barrier to entry is not musical aptitude but willingness to put up with bullshit, because it feels like bullshit when you are doing it.
That sounds like a the ear training I have done but even after a couple of decades I don't have any real ability to tell the difference between intervals. For example, I listen to a musical interval and try singing up the major scale to match it but everything matches a fifth (in my internal melody). More recently I found the Sonofield app an interesting idea for learning the intervals but wasn't able to gain any consistency.
I remember many years ago in my music lessons being shocked that some people can hear multiple notes played simultaneously. I've never found much material on learning this skill.