Achievements are important. Ever since their inception during the Xbox 360 they invented a new way to play games. Achievements get people to give new modes a try, it gives people a goal to strive for when they completed the game to give it more replayability. It gives you an excuse to try new games because they got a easy set of achievements to do to add to your collection.
It's why Trophy Hunting is a big thing on the PlayStation side. I can't imagine a world where achievements ceased to be because of the additional fun they offer. Just ask major PlayStation channel MysticRyan and his big collection of Platinum Trophies.
But you can't just have achievements. If you implement achievements it needs to be a major feature, not tacked on like Apple Game Center ones are. You need a way to brag about your achievements, a profile score that goes up the more achievements you do. Just simply having achievements doesn't make them enticing. For PlayStation it's Platinum Trophies, a special trophy you get for getting all the trophies a game has to offer. For Xbox it's your gamerscore. For Steam it's a completionism marketing as well as a rare achievement marker, and the achievement showcase on your profile