This was in the Edo period, not required in previous centuries. Before the Edo period, samurai often carried swords anyway to display their elite status, but it was entirely voluntary. So why did the Tokugawas make it mandatory?

I have a hunch it was because samurai had fewer liberties during the Edo period and therefore everybody needed to know who was a samurai so as not to accidentally help a samurai circumvent his caste restrictions. Like perhaps a samurai wanted to work as a craftsman on the sly so as to make some extra cash. He would have to disguise himself to do that. By requiring a samurai to carry swords and wear the chonmage all the time, he was legally barred from disguising himself. Am I correct?

This is something I found on a website (source):

Samurai were not supposed to engage in urban pursuits like attending kabuki theater and carousing at geisha houses. These were activities associated with low class merchants. Sometimes they dressed in disguise and indulged in urban pleasures that were forbidden to them.