The Chiyuv of וקדשתם or of giving honor doesn’t translate right into bring superior. A king is obviously special in his position, but there are Mitzvos that are specifically set to make sure that בלתי רם לבבו.
I haven’t ever seen that a Kohen has a higher Neshama. On the contrary, it has been mentioned that people are נתגלגל as one in order to end up with all the commandments. So obviously it’s not because of the Neshama. It is an honor to Aharon and his progeny, and the Kohen is tasked with special duties and therefore deserves certain honors. If this causes him to be proud of himself, he is silly. And it’s embarrassing to be proud of what you didn’t accomplish.
However, he is indeed born into a special position that he should realize is special, and make sure not to ruin the honor of that position. The same goes for any special bloodline, or Yichus.