For sure those two characteristics of the feedback system (step response of the input and step inside the loop) are related. And yes, by stepping the input we can see if the system oscillates. My only purpose when advising for injecting a disturbance inside the loop was to see how the loop corrects for that and how it does it.
Junus, it will be difficult to come with a compensation strategy online. But it looks to me it won't be as easy as just placing caps to ground here and there. You have 3 stages in your CMFB loop - the first stage, the second stage and your error amplifier.
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Also, don't forget the local feedback loops of the cascode compensation (I think we discussed that some time back). Have you tried placing caps in parallel with each of your output common mode sensing resistors? This will create a zero in the CMFB loop.
Since you're simulating with iprobe, you get the overall loop gain as a result. A good way to debug these loops will be to go to the old way of simulating ac stability. For example, break the loop between the inverting input of your error amplifier and the middle point of the common mode sensing resistors. You break it by placing a huge inductor in series - 1 to 10H. It will keep the loop closed for DC but will break it starting from some very low frequency. At that break point, on the side of the error amplifier inverting input connect a big capacitor (for ex. 1F) in series with your test ac voltage source (dc =0 and ac=1), the other side of that voltage source connected to ground. Run ac (not stb) simulation. At least for low frequencies you should see the same loop gain as with iprobe. For high frequencies there will be differences and also now you'll have a resonance peak at some low frequency (hopefully very low) from the big L and C. With this setup you disturb the loading of the loop at the break point but if it happens at sufficiently high frequency, you'll still be able to debug at lower frequencies.
Now, this setup allows you to find the transfer function between any two points around the loop and see where the troubles come from. For example you can look at the transfer function from the input of the error amplifier to its output. Then you can look from the output of the error amplifier to the output of the 1st stage of the main amplifier and so on.