LDO is a feedback system, and it uses negative feedback to set the output.
It can happen, that by the parasitics of the transistors this negative feedback turns to positive, and the whole circuit will oscillate. Without compensation and with more stages you increase the risk that it will happen.
I don't think most of the LDOs use 3 stages, when regularly 2 enough, and in most of the 3 stage LDOs the 3rd stage is not a common source amplifier, usually it is a source follower, because it doesn't add low frequency pole to the system, or the 3rd stage is not in the feedback loop, just a current mirror output for a load.
Your design at least have got 3 poles, and 2 are close to each other. The open-loop gain also very high in this circuit if all transistors are in saturation. These means the phase margin is small, and because of the 3rd pole probably this system's phase margin is negative, and it is unstable.