It may not be as important to you. You may think City winning the premier league should be more important to a player than his country winning the World Cup or European Championships. I honestly don't think that is the case with the players.
Footballers have, from their point of view, lamentably short careers that are a single bad tackle away from being brutally terminated.
They cannot be criticised for trying to squeeze every last drop from the experience. To fail to appreciate this, is to fail to understand human nature.
I appreciate that, but you must appreciate that precisely because they are one bad tackle away from it all being over, that they should probably pay a little more heed as they get older to what's best for their long term as well as short term future. Of course going to play at the Euros is a great experience for Silva. But baring in mind he's won a world cup and two euros already for Spain, but has no CL winners medal, there's surely some sense in trying to achieve what he hasn't achieved and prolonging his club career rather than going to experience a tournament he's won twice.
I don't think that we can use broad brushstrokes and say that either or is important to all players. My point is simply that no matter what personal satisfaction they take from international success, their job is with the clubs they are contracted to. That's who pays the wages, gives the platform for international recognition etc. So players in general should show a little more respect to their clubs.