I will completely withhold my opinions on de-beaking and will try to present only the science of it:
I study avian biology. When "Properly" de-beaked, the fact is that the knife must cut into the base of the beak, where the cells that re-grow the beak rest. This is to prevent the beak from simply re-growing.
This area is full of nerves and nociceptors (pain receptors). It has also been studied that it is not unusual for neuromas for form in the beak as it attempts to heal... while we do not know exactly how these make a chicken feel (truly, the only way to know is to be a chicken), neuromoas in humans can cause chronic pain where they form.
Nerves run nearly to the tip of a chicken's beak, and nociceprors have been found down the length of the nerves.
In more personal observation, I have a hen that has broken her beak several times-- she's terrible about getting into weird situations-- and even when it is just the tip, I have noticed that it causes her pain to eat until it has healed.
De-beaking factually does cause pain, and speculatively might cause chronic pain.