5ed does a poor job here - the Paladin blurb is "anti Evil" but there is nothing saying that I can't be a Paladin of Vengence, my particular beef being against the Seldarine in general - for chasing grandma into the Underdark, or Correlon Larethian in particular - for taking my god's eye out.
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I would argue that that's a feature, not a bug. 5e is clearly presented as wanting to provide a lot of openness and freedom for players.
I am fine with Paladins from outside the top left corner.
My beef is the disconnect between the PHB flavour text and rules:
"Whatever their origin and their mission, paladins are
united by their oaths to stand against the forces of evil."
SNIP
A paladin swears to uphold justice and righteousness,
to stand with the good things o f the world against the
encroaching darkness, and to hunt the forces of evil
wherever they lurk."
SNIP
"Fighters are rare enough among the ranks o f the
militias and armies o f the world, but even fewer people
can claim the true calling of a paladin. When they do
receive the call, these warriors turn from their former
occupations and take up arms to fight evil."
SNIP
"Evil lurks in dungeons and primeval forests,
and even the smallest victory against it can tilt the
cosmic balance away from oblivion."
If what you described is what a player wants to play, and everyone at the table is okay with it, then what's the issue? That's a very interesting and fun story potentially.
Paper and pencil game? No problem. Now put that into code for a cRPG ...