What I think you meant to say is that Latin does not use pronouns in its verbal conjugations. That's a feature of most ancient Indo-European languages. It did, of course, have a full suite of pronouns used in other contexts.
Oh, and no, the plural wasn't used as a formal singular. It was idiomatic to occasionally use the first person plural to refer to oneself (you can see this in Cicero's letters), but it had nothing to do with formality or some sort of "royal we."
Wait, I definitely learned of the royal "we" in Latin. Specifically, this came up in Apollo and Daphne[1], with much of Apollo's first speech in the first-person plural referring exclusively to himself. We were taught in class that it was related to his arrogance to use a royal "we", which is a large theme of the story.