His occupation is listed as sellator, he says, which makes sense on a charioteering team; sella means "saddle", and one would think that such a man would be in charge of the tack, correct?

No, in fact. Some scholars believe it is a spelling mistake, perhaps an instance of someone mishearing, and it really being cella, which means "room", so a cellator would care for a storeroom or a series of storerooms. This is due to the fact the only other instance where sella appears in surviving Latin texts is in the word sellarius.

He scribbles the word on the blackboard and circles it with a jaunty flourish.

A sellarius is a male prostitute -- or in other words, the "saddle" you ride while you screw his ass.