There’s nothing wrong with chewing gum on the street – just make sure that you crawl on all fours and say moo.
Otherwise – as Harav Menashe Klien Shlit”a said (he was being asked about chewing powder gum on shabbos – which many poskim have said is an issur de’oiraisa of Losh), “Is such a thing muttar during the week?!?”
It is simply not honorable for a human being to do it.
They used to say (in the days when it was applicable) if it’s too demeaning for the president to do in public then a Jew is definitely far beyond such things.
That’s also why it’s not tznius. Tznius is about being a tzelem elokim above all the other creatures of the earth. True, we must eat – but we don’t need to chew our cud.