If Euler angles have so many problems, why do people still use them?
Well, for one, they're much easier to understand and wrap your head around :). Actual answer - In the real world, they're directly measurable via sensors like the gyroscope. As far as I know, there's no sensor that can directly tell you the axis you're rotating around. Wikipedia has a section on other applications if you're interested: https://en.wikipedia.org/wiki/Euler_angles#Applications