Basically, for any given opamp there is a fixed time delay between a signal being input and arriving at the output. This is mainly due to R/C time constants in the signal path, created by parasitic capacitance in transistor junctions etc.
Mostly we want less than the max gain, so we apply negative feedback.
As signal frequency increases, the inherent delay becomes a larger proportion of a signal cycle. When it exceeds half a cycle, the negative feedback becomes positive feedback.. and all hell breaks loose.
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As to working out what values to use, sorry but that does require all those funny squiggles. Or else just try and see. Which I suspect is what's used in most cases. Square wave testing will typically show ringing or overshoots if the stage is undercompensated.
A good discussion of the problems related to poles moving as a function of frequency. Circuits that are stable at DC can go unstable as the frequency is increased.
I think those squiggles are the difference between engineering and technicianing (I know it's not a word!). Most feedback systems are not simple op amp circuits, we can't plunk down feedback capacitors. Lots of times there are hydraulics, motors, valves, temperature controls and other physical phenomenon that resist tinkering. The frequencies are lower but the lag is considerable. Just getting a PID loop correct can be a lot of knob twisting (not withstanding 'Autotune').
In school, we had two courses related to controls. One instructor worked on the controls for the rocket nozzles on the Atlas missile and the other worked designing controls for nuclear power plants. It isn't as easy as you might think to get the Atlas away from the launch pad without falling over.
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