So the question is, when do you find yourselves shooting in manual, or what situations require it?
I think the key point is that with manual, you
are in control. If you preselect an aperture or shutter speed and just adjust the shutter speed to get the metered exposure, you're effectively doing what Av mode does, only slower (and vice versa for Tv mode). So why do it at all, then? Because if you start out doing it that way, it becomes easy to decide you want the image a little brighter or a little darker, and adjust on the fly for better (i.e. more desirable to you) results.
Could you just get in the habit of applying EC routinely when needed? Sure, that would work, too. But M gives you the flexibility on how you apply EC - e.g. your subject starts moving facter so you want to increase shutter speed, vs. your subject goes into shadow so you want to open up the aperture a stop - in those cases, if you had to switch the camera from Av to Tv, you'd miss the shot, whereas the decision to change the exposure with your thumb vs. your index finger is a lot faster, and with practice it becomes automatic.