A convert can marry a Jewish king. A convert can marry a prophet. A convert can even marry a rabbi. So there must be some reason for a Kohen not being allowed to marry a Kohen.
Here is Kabalistic thought:
When the Torah forbids a marriage, it is never because one party is not good enough for the other. It is because both parties are not matched to each other. They are simply not soulmates. In the case of the Kohen and the convert, their soul dynamics clash, their spiritual energies contradict, and so they can’t marry.
The holiness of a Kohen is hereditary. If your father is a Kohen, then you are a Kohen. Priesthood is a birthright that is not achieved through a person’s effort nor deserved through a person’s righteousness. It is bestowed at birth.
The holiness of a convert is the exact opposite. It is completely earned. The convert was not born Jewish. They chose it. They achieve Jewishness of their own initiative and with their own hard work. They are self-made souls.
So these two souls, the kohen and the convert, are moving in opposite ways. The kohen receives their power from above. The convert creates their own soul energy from below. The Kohen has the ability to bring down blessings to others, just as their soul was given to them as a blessing. The convert has the power of innovation, of initiative, of creating holiness from the ground up. They are going in different directions. For this reason, their souls are not a match.
Both the Kohen and the convert have awesome holiness. It is a great privilege to be gifted with the soul of a Kohen. And yet, the self-made soul of a convert has a depth of experience that inherited holiness cannot compete with. Neither are second class souls.
The Kohen is crowned with a legacy from past generations. A convert creates their own legacy for future generations. The Jewish people is richer for both of them.
Mendel (Menachem) Bluming
Sources: Shaar Hagilgulim 34:3-4 of Rabbi Isaac Luriah and other sources