Definitely. There is a right and a wrong way to do variety.

I think that on a psychological level, too much meaningless variety tends to stress people out. The best example is the toothpaste aisle in your local supermarket. I've seen a lot of studies on this in recent years highlighting the fact that the sheer amount of meaningless choice our society produces is detrimental to people on a psychological level. This is exemplified in the marketing/branding strategies employed by a lot of tech giants like Google and Apple: these companies pride themselves on offering simple, accessible services that cut through the layers of bullshit in  modern life. It shows in every facet of their business model to the functionality of their services to the colour schemes to the design of their products and so on. Most humans get stressed out when presented with too many options, we suck at multitasking and we are all suffering the effects of information overload: companies like Google and Apple know this and the simple solutions they offer are alluring.

Unfortunately, I don't think most AAA game developers have learned this lesson. The reason being that some humans, ie: the types that become cash "whales" in MMOs, actually like hording and don't mind superficial variety. There is an evolutionary basis for that. Most publishers and developers alike saw dollar signs in whales their obsessive hording and based their business models around it because it's possible, despite the fact that most gamers don't fall into this category. So what we have now is this business model becoming diffused and spread across many different genres of games, even genres that were historically very simple on a superficial level but very deep in terms of mechanics (FPS). Then there is the fact that it's easier to make games that are wide and shallow than it is to make games that are narrow and deep: it's easier to market them ("look at all the stuff we've put in here!") and it's easier to make them (just have artists pump out more and more of the same on the game development equivalent of an assembly line). 

I think that the majority of gamers (and humans) actually prefer the "simple to learn, difficult to master" approach for games. It's elegant. That's why things like Rocket League and certain Indie games like Super Meat Boy or Flappy Bird go viral: they're painfully simple to learn but mastery is a different thing altogether. I think we tend to be put off by meaningless variety but not by meaningful variety coupled with lots of depth.

The worst offenders, in my opinion, are things like the modern Battlefield games, Ubisoft's open world games, and MMOs that base their business models around catching whales. That's why games like this are often called "Skinner Boxes" as they rely on superficial variety and instant gratification as a substite for meaningful depth. It's depressing how Ubisoft pumps out gorgeous sandboxes like 18th century Paris or modern Chicago and then does absolutely nothing interesting with it