Matthew, your arguments equivocate on the meaning of 'secular'. Two definitions apply: 'not an ordained clergyperson', and 'not concerned with deity or religion'.
You state (correctly) that a paladin is secular, definition A. You then argue that they don't draw power from a god, or even need to serve a god at all, because they're secular -- but that would be secular, definition B.
Paladins get divine spells, divine grace, divine health, cleric-like class features... they are affiliated with 'divinity', whatever that means in a particular setting, and thus they are not definition-B secular, even if they are not ordained clergy.