Yes, "It would be a great mistake to argue that Art and Music should be optional/should not be compulsory subjects" is natural English. In the United States optional courses are called "electives" so one might say "It would be a great mistake to argue that Art and Music should be electives."
"Argue" has a range of meanings. It can refer to a spirited but polite intellectual discussion. "They liked to argue for hours about Keynesian versus Hayekian economics."
A single person can "argue X," or present an "argument for X," which means a civil, polite presentation that gives reasons to believe that X is true.
There is an adjective, "arguably," which means that the speaker doesn't believe something but acknowledges that it could be true. "Arguably, Pluto should be considered a planet after all."
There is also a second range of meanings in which an "argument" means an angry dispute, a row, an altercation, a quarrel. "We were enjoying our dinner when the couple at the table next to us suddenly began to have a loud, heated argument. It went on and on until eventually the maitre d' asked them to leave."