Daemons aren't really physical beings. They are the essence of something. They are dreams and passions given life as spiritual creatures. They might wear a physical form for a little while when they enter real space, but that body isn't really who they are.
Angron can't lose his nails because they're a part of who he is. They are a symbol of what it means to be Angron. It's the same reason Mortarion still wears a gas mask, or Perturabo still has armor.
He might even still hate the nails and want them removed. But that's just because hating the nails is part of who he is.
Sometimes we'll see a daemon permanently changed. They'll be scarred by a wound or lose a possession. But those changes are always heavy with symbolism. Just killing a daemon isn't enough to cause those changes. It comes about when they are defeated by a holy champion, or trade away some power for influence. Maybe Vashtor/Angron/Khorne could remove those nails, but it would represent a fundamental change to the core of Angron as an idea.