It kind of creates a psychological effect as you get immediate satisfaction/gratification for doing a specific thing in a video game. As for me, they aren't really mandatory, but if I see an achievement or trophy that looks reasonable to accomplish I may try to go a bit out of may way to get it.
Well I have a different mindset. In my opinion this satisfaction you speak of exists, yes, but it's only artificial. A game should be entertaining enough on its own to be considered a good game. If the satisfaction of playing said game only derives from fulfilling achievements, it isn't a good game. On the other hand, a truly good game doesn't need achievements because it' entertaining enough. I know that in this day and age I may be the minority in this regard.
Also, I believe Nintendo will bring it's own achievement system to the Switch one day or another. There are many games in their portfolio that use kind of their very own achievement systems already (like the buttons in Arms, for example).
Gameplay > Graphics
Substance > Style
Art Direction > Realism