There almost was a King Arthur, though not the King Arthur everyone thinks of. Arthur was the older brother of King Henry VIII, but died before becoming King, hence Henry VIII.
Before that, the brother between Kings Richard I and John was Geoffrey, Duke of Brittany. His son, Arthur, was denied his immediate right to rule by Richard and was likely killed in exile by his good uncle, John.
Of course, Arthur of Brittany was conspiring with Philip II of France to take the British throne, but he was still the rightful heir.