You can get so into the technical stuff you can even arrive at an incorrect conclusion like something is not useful. From a practical standpoint everything is useful. Channel Tools is a very useful plugin to have around. As someone who mixes a lot I tend not to think of it as a device for increasing apparent width but much more often for me anyway decreasing width. It is also good for fine tuning levels of both L and R channels.
I am working on a mix now that is big and complex and quite a lot going on right across the stereo field. The client supplied some extra pad like tracks. They were nice musically but very wide. Due mainly I would so say to the processing that was on the last stages of either a virtual or hardware instrument.
(and also in programming of course) When put into my existing mix I felt it was just out on the sides too much. I wanted some stereo but not so wide and I wanted that whole sound to move to the left slightly. Channel Tools did an excellent job. Once it got out of the way of the sides and moved to the left a little it became easier to hear. Maximum illusion, minimum voltage. I was able to turn it down.
I got a bass part that was also very wide
(some wild panning) and it too did not sit well being so wide with a wide mix so I planted both channels in the centre and it too became clearer and louder, able to turn it down and still hear it etc..
I have also had the situation where I either recorded or received a nice pad part in a piece. It was stereo and sounded nice at the time. On further listening I discovered it was not stereo enough. It had a little width. It was having trouble being heard. There are better processors than Channel Tools out there that will widen a sound like this and put it well into the sides and well away from the centre. And doing it all without using delays but much more interesting techniques that you don't need to understand. All you need is to understand what they do to the sound. What a good widener will do will make the pad sit well out away from the centre so it becomes very clear and easy to hear. I was able to turn it down again.
I check all mono compatibility by summing L+R and listening to a small single point source mono speaker. The better more interesting widening processors play well with the mono check. Once you start turning the polarity of one side of a stereo image upside down is when the mono test fails badly. eg some parts disappear all together! That is a horrible effect in most circumstances. I prefer to leave that alone.
It is also very handy to be able to flip the polartiy of one side of a stereo signal too. I have used that correct a bad stereo effect that must have used the same technique. And it is also nice to be able to use the delays on one side. For a slow moving pad I have been known to add 100 mS or more to one side. It creates a lot of width due to the time distortion going on. And it sums great in mono too.