The "sake of his career" argument might make slightly more sense if he was 5 years younger but he's 29 next. There's probably 2 major England tournaments left to aim for but what chance does he really have of re-establishing himself for those?
If he moves to a club lower down the league he won't be competing for trophies. He MIGHT rediscover his best form over a couple of seasons but he's 30-31 by that point, so what big club is going to come along and give him his big chance then?
Sure he earned a bigger salary than most people earlier in his career but he was a championship player mostly. It's only over the past couple of years where he's been earning real megabucks i.e. the kind of money where he can retire and look after all of his family for the rest of their lives. The guy grew up dirt poor and he doesn't owe it to anyone to give up at least £100k a week. From his perspective he's probably willing to keep fighting for his place.
Fans are first to kick off when a players leaves to earn more elsewhere, but will then readily attack a player when they think they should accept being shipped off for not meeting their expectations.