Incorrect. Oathbreaker is assumed to be evil. They most likely are as "most" Paladins are initially good but not all...and even if they were good breaking their Oath doesn't inherently make them evil.
Which point are you trying to make? I literally said breaking an oath does not make a paladin evil, however paladins of the oathbreaker sub-class in 5th edition are exclusively evil.
5th Edition Dungeon Master's Guide page 97: "An Oathbreaker is a paladin who breaks his or her sacred oaths to pursue some dark ambition or serve an evil power. Whatever light burned in the paladin’s heart has been extinguished. Only darkness remains."
It's not the breaking of the oath that makes them the oathbreaker, it's the pursuing "some dark ambitition or [serving] an evil power". And if that's not specific enough for you, 2nd paragraph states:
"A paladin must be evil and at least 3rd level to become an Oathbreaker. The paladin replaces the features specific to his or her Sacred Oath with Oathbreaker features."
Secondly....Death Knight is an undead entity and not something you become by going down an evil Path as a Paladin.
5th Edition Monster Manual page 47: "When a paladin that falls from grace dies without seeking atonement, dark powers can transform the once-mortal knight into a hateful undead creature."
Also look at the powers of the oathbreaker sub-class; controlling undead is very much on course towards the death knight's skillset, and death knights are unambiguously evil.
Please don't say "strictly speaking" when talking about actual lore topics without being properly informed on the topic. You are spreading opinions or assumptions only partially informed and not completely to the actual lore of D&D and it at best does nothing and at worst hurts the hobby.
Don't accuse people of being incorrect or spreading falsehoods when you can't even be bothered to check what they've said; I'm entirely 100% correct in this case.
The oathbreaker paladin subclass is literally listed in the DMG as a villainous class option; it's specifically intended for evil paladins, there is no hint of ambiguity whatsoever, and it's not even intended for normal use by players at all, it's for NPCs. It is only a player option with a DM's explicit approval, because it has interactions that can actually bolster enemies and make a campaign harder for the players, though the BG3 version lacks these.
What it isn't, is a sub-class for every paladin that breaks their oath for whatever reason. This is why it's always been a stupid name for the sub-class, because it only confuses people into thinking all paladins that break an oath must become an "oathbreaker" paladin, but they don't. It should be named "Apostate" or something more suitable.
It's also worth adding that having an evil alignment doesn't necessarily make a character murderous, it makes them ruthless, usually arrogant and selfish; an evil character can do the right thing for the wrong reasons. Mercenaries and thieves can be evil if they don't care who they fight/steal from. But oathbreaker paladins specifically have "dark ambition" or "serve an evil power" which implies a step beyond that.
Paladins who break their oaths are paladins without oath-specific powers, but choosing a new oath, or re-speccing as fighters or something else are entirely valid paths for them to continue in 5e without being forced to either renew their oath or pursue a "dark ambition".
The oathbreaker himself (who appears in camp) shouldn't be an oathbreaker paladin either; he should have simply switched from crown to vengeance or something else more appropriate since he could no longer follow his king.