Yep, it's fine to roleplay a character who can't be cured, or just prefers it that way. Just as it's also fine to have a tabaxi be a vegan, because it's a fantasy and players are allowed to express themselves however they wish and the lore is always secondary to player enjoyment.

Even in a setting where having your head chopped off is a minor inconvenience for anyone who has 1000g, there would still be people who choose not to have an amputation healed for any number of reasons, and thay choice should be up to them.