Why can't a lefty play third base?

They somewhat can, it's just not as convenient. When playing on either line, it's just an advantage to have your glove on the opposite hand of which line you're on so that you are at a better angle to field the ball. Think about it, if a lefty 3rd baseman is trying to field a ball to his right, he will have to back hand the ball or cross over his body to catch it. it's easier to field a ball to your glove side, and when playing on a line you wouldn't need to field many balls on that side because they will likely go foul. So it's mainly for fielding purposes rather than throwing or tagging.