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The Round Table represents everything that separated Arthur from Uther. People repeatedly point out how Arthur isn't so different from Uther as he'd like to think, because he, too, shunned magic. But that is because of upbringing, and inputted thoughts.
Arthur believed in equality.
The first time he met Percival, he told him to call him Arthur instead of the expected address (my lord, sire, my king). When he first found the Round Table, Merlin was an inseparable part of it, even though he was his servant. This is because he remembered and agreed with what Gwaine said, "Nobility is defined by what you do, not by who you are." He married the servant girl, daughter of a condemned blacksmith, for goodness' sake, because she was wiser than most nobles.
Why he refused to accept sorcerers and magicians is because he did not see them as equal. Every single sorcerer he had met had betrayed him in one way or another, were dangerous people he needed to be careful around. Nimueh, the witch-hunter, Kara, Dragoon the Great, Mordred and his beloved Merlin. His mother and father both died of sorcery. He lost his half-sister to magic. They were criminals. Why would he allow sorcerers equal status if he wouldn't grant that to petty criminals or those acquitted of treason?
No, Arthur was fair to the very end.
He promised Dolma that he would remember there is no evil in sorcery, only in the hearts of men, and he forgave Merlin.