It's fairly simple. PM is and has always been a competitor. This is undoubtedly his last RC.
So, he's not gonna bow out. And he probably gave Furyk an earful today when he learned he wouldn't be playing either am or pm Saturday sessions, because he's a competitor. And even though he didn't earn the right to represent his team today, I would respect him more for fighting to be out there in one of the Saturday sessions than "recusing" himself. It's the nature of highly competitive "competitors" who've achieved in their careers.
It's not like we're talking about Shaun Micheel here.
And if anyone on this current Ryder Cup team, from either side (besides Tiger) achieves more than Mickelson has, on the world's biggest and most competitive stage (9th in all-time wins, 5 major victories) - I'll gladly apologize on his behalf. π
That said, he needs to win an f-ing point tomorrow in the singles.