I feel like this is being made too complicated.
Magic isn’t push button. Just because a paladin gains powers from their oath and conviction doesn’t mean that everyone who swears an oath with conviction becomes a paladin.
Why does it work that way? Hell if I know. Maybe it’s a decision of the universe. Maybe there’s a god of oaths who decides who has true enough conviction to be granted paladin powers. Maybe there’s an organization who trains knights to be able to become paladins. Maybe there’s a tribunal of angels who assigns an angel to oversee every nascent paladin to see if they’re worthy.
It’s up to you as the DM, or even better, work with the paladin player to decide. If D&D was an actual setting, the rules would probably tell you. But D&D is a toolkit, so you’re going to have to do the heavy cosmological lifting yourself if you care enough to look for an answer.