Bad Things Happen - to the characters. Not the players.
When a player that has caused a problem says, "But that's what my character would do!" we generally find that to be a poor excuse. From a game design standpoint, "But Bad Things Happen!" is just, "But, that's what my world wants to do!", and is a similarly poor excuse for giving a player a bad play experience by design.
Which is why a lot of modern games take that out of the design. There are ways to make bad things happen without screwing over the player for long periods of time. Indeed, in the best designs, a bad thing happening to the character is a cool and interesting event for the player. Not a boring one.
If the intent of the design is, "Get hit and fail your save, and you LOSE," then maybe they should just die, yes. After all, that's really the basic intent - it just lacks the courage of its convictions to actually do it.
If the intent of the design is, "Getting hit and failing your save complicates your life, and you should avoid it," there's several different ways to implement that. Maybe try one that doesn't mean the player is personally better off just leaving the table.
In my opinion and approach to gaming, if an established player would be better served leaving your table, as a GM, that should be considered a failure mode.