Yeah, I was on reddit earlier and someone posted this quote from the phb:
“a paladin’s power comes as much from a commitment to justice itself as it does from a god"
Apparently the SCAG specifically points out the connection between gods and paladins.
Lastly, Ed Greenwood mentioned how atonement works, and also mentioned the phb:
"If a paladin transgresses against their oath, the usual absolution, as the PHB states, is to seek absolution from a cleric of the same faith. "(emphasis mine)
I don't have access to the SCAG or PHB right now, so someone else would have to verify this, but it's pretty clear to me that deities and paladins still have a strong connection in 5e Forgotten Realms, and that the oft-repeated claim that 'in 5e (Forgotten Realms) you don't need a deity anymore to be a paladin is some sort of fanon/misreading that's just been repeated enough times that people just take it as factual without looking into it.
I do I have PHB, the "breaking your oath" section full quote is: "
A paladin who has broken a vow typically seeks absolution from a cleric who share his or her faith or from another paladin of the same order".
I do not know if there are non-Faith based Paladin Orders in the Forgotten Realms, since the one I've seen have at least one "patron" deity. And absolution is the forgiveness of sins...
That section also give examples that cause broken oaths: "
Sometimes a situation calls for the lesser of two evils, and sometimes the heat of the emotion cause a paladin to transgress his or her oath". Oath breaking should be pretty commons, but for small things you just go to the confessional to clear it up. Talking to the Oathbreaker Knight fills that role in the game, since he ask you why you did it and you can pay him to atone.
I'm not quite sure if the belief is a fanon telephone thing. I think WoTC promoted it that way officially and their words were given more weight than what they wrote in the PHB/DMG. Also, in other official supplements they released Paladin subclasses that fits their no alignment/no god idea better. But some of them don't really feel like Paladin to me, they just happen to use the class structure for abilities.
They could have named the class Warlord or just plain Knight if what they wanted to do was a non-divine support warrior class too.
edit:
I actually think Larian should go homebrew with the oaths and put gods back in:
Devotion is fine as the classical justice/honor/goody-two-shoes Paladin. Meant for good/neutral gods.
Of the Ancient should be purely nature themed instead of the weird nature abilities but good/beauty/art oath tenets it has in the PHB. Meant for nature related gods.
Vengeance should allow the player to select their sworn enemy/enemies (because "greater evil" is rather vague). Works for any gods.