In the kingdom, the Yonkous are those who possess the flame of life and purpose, they don't care if they die to achieve their goals. They plunge into this vast sea headfirst to fulfill their ideals and beliefs, to accomplish what suits them, unlike the admirals who follow the orders of their superiors despite believing in something. They all have gigantic ambitions and cling to them fiercely.
Conquer, kill, destroy, enslave, steal, manipulate they are willing to do anything to achieve their ideals. Among all the Yonkous, Kaidou is the one with the greatest purposes for the world, he truly wants to radically change it.
The justice of the admirals, when compared to this, is not enough to earn them the COC (Conqueror's Haki) and the will to be king. Anyone can carry a sense of justice within themselves, even an ordinary citizen. However, not everyone desires to conquer the world, destroy it, and so forth. The Yonkous' ambitions go far beyond what mere justice can encapsulate.
In summary, justice does not bring with it the desire to conquer or to be above everyone (or something specific). It rather represents a feeling of bringing about change, though not necessarily on a grand scale. The Yonkous, on the other hand, have ambitions that go far beyond a simple sense of justice and involve pursuing immense power and dominance over the world. Their goals and aspirations extend far beyond what justice alone would typically encompass.