Wait? I never heard about the management wanted to drop Button but the board didn't. Even if the management wanted to drop Button, that doesn't mean most of the people at McLaren racing didn't want Button. It might just mean that one person... yes one person only (which might be Ron Dennis) that wanted to drop Button. And it probably not because he disliked Button, but more about the potential for Magnussen as he is a lot younger than Button + he is cheap + he might attract new sponsors.
If you asked any team, and money doesn't matter, which one would you choose... Alo - But, Alo - Mag, But - Mag. I think the answer is obvious. And it's more obvious for a top team that wanted to win (which is basically choose the best driver available for now and not for the future).
So I don't know if they are going to race in China with the feeling that management don't want Button to be there, but I haven't felt that. Not China, not in Malaysia, not in Aussie, and not in testing. So whatever the management problem with Button is, it certainly didn't affect their race operation. Which is fine by me
Should he retired? Why should he? The reasoning in that article didn't make any sense. He might not need to prove anything, but then why Alonso still want to race? How about Hamilton, Kimi, Vettel? And the fact that Button facing someone considered to be currently the best racing driver, I think that should motivate Button enough.