Okay, to clarify, tow the line would work perfectly well in a nautical sense: A sailor flings rope to a receiver on land who physically tugs, pulls, and tows the line attached to the boat until the vessel is safely docked and secured. It’s a metaphor for “working hard.” Lovely, but not what the phrase means.
Instead, the true phrase is toe the line. Yes, imagine your little tootsies kissing a line on the ground. The expression comes from an old practice common in schools, the military, and other institutions where daily roll-call was the norm. Individuals had to toe the line, or ensure they were perfectly lined up, standing at attention to yell “Here!” or receive orders. The original phrase is about obeying the rules.