A narrower definition of "useful character" in this discussion would be helpful.
dictionary.com:
useful (adj.) [yoos-fuh l]
1. being of use or service; serving some purpose; advantageous, helpful, or of good effect: a useful member of society.
2. of practical use, as for doing work; producing material results; supplying common needs: the useful arts; useful work.
I'm thinking we need to be in tune more with definition (1.) than (2.). That is, a character represents an imaginary person who should be useful in the sense of "serving some purpose" rather than just a tool "that is of practical use for doing work"
A lot depends on game and setting: in some games, if a character is part of a team, that character may be more specialized to round out a needed set of skills, attributes, and abilities within the team. If the setting permits more individualized play, a player may wish to try out a specific set of skills, attributes, and abilities - a role they haven't played before, or that player may wish to play a more familiar, more comfortable role in the game. This kind of stuff goes beyond the rule system, and is more a matter of how the GM and players interact.
From my perspective, the only character that is not useful is the one that is not playable and enjoyable to the player.
A character can even serve a useful role in death if it's done correctly. This is where the influence of the GM makes all the difference. One of my mentors, back in the mists of antiquity, once told me that the way to know when a character's death is handled correctly is when the player walks away feeling like they got a fair shake, and the character's death served -
wait for it - a useful purpose in the game to that point.
