Big name chess players do more than play chess. Journalists ask silly questions. Fans want autographs and selfies.
Do you think the players like or just tolerate these "extra duties"?
Im pretty sure they get paid to do that, through sponsorship deals etc
You're right, as soon as the game is finished the players should run off to their room, lock themselves in and turn the engine on. That will help increase the popularity of the game.
Signing autographs and granting interviews are great ways to promote the game of chess. Running away after every game will just make chess more boring than the way non-chess players perceive it to be. So, whether they are being or not, they should sign and do whatever it takes to promote chess.
As in many other areas: some love the attention some find it annoying. And, likely, most are somewhere in between.