I didn't watch the vid as I seriously detest theories in a video format. The least you could do is giving a summarization of the vid and add your own input.
Either way it's not unreasonable to assume that marines like the admirals have the King's Disposition, but that they simply haven't shown it yet. If you're not going to assume that, well it can be explained by the reality that the world of pirates is a much better environment for the KD to prosper in contrast to the world of the marines. Someone who has the KD automatically attracts other people around him and is destined to stand above others almost like it are laws of nature. This makes the life of a pirate a good match as pirates travel around the world unrestrained, not bound by anyone or anything. The marines however have a hierarchal structure and everyone has to listen to their superiors, they can't go wherever they want to go. Someone who inherently doesn't easily accept authority, is really not going to join an organization like the Marines.
If you want to visualize this, none of the persons that currently possess the KD are the type that are going to accept someone else's authority easily. DD and Hancock, despite being Shichibukai, gave the Marines the finger right in their faces. Akainu on the other hand was forced to follow whatever game the WG was playing with DD.
Most users of the KD wouldn't really be interested in joining the Marines.
Doesn't Cobi have Conquerors?
He doesn't. In the least there are no signs he has it. Cobi knows Haki though and has an affinity for Observation Haki.