How long before people stopped being little crybabies?
And it's a perfect metaphor, these people who think they are so good that they should never lose.
Big Head Mode wouldn't work. If I quit and got that penalty I would just idle in the next game to reset to regular mode. So rather than solve the quitting problem, you've just greatly increased the idling problem.
The easiest way to stop quitters is to focus on making the gameplay fun while managing the playlists in a way that prevents players from having to play a game type that they don't like.
No more Snipers games outside of the Snipers playlist. No Race games outside of the Race playlist. Don't ship maps with easily exploitable features (original Zealot for example).
Don't design a
shooter with gameplay that disproportionately rewards players who go for kills by doing things other than shoot (grenades, wacky AA's, punch-punch).
No more four-way split-screeners vs. a team of four full-screeners.
Ranked play to discourage quitters for the competitive players and drop-in/drop-out for social games so that you don't get the quitter cascade effect.
Do ALL of this and you will hopefully cut quitting to half the rate we see in Reach. The quitters will always be with us, but they don't have to overwhelm us.