Design should be focused on the user, not the design itself. It has always bothered me when people working on a creative project (whether directing, managing, designing, providing feedback, etc.) are more focused on their personal pride in the design than on the purpose of the design. In the case of a website, that purpose is usually to allow a reader to get information, interact, and accomplish some task.
I have nothing against flat website design in general, but when taken too far and applied in the wrong way it competes with the usability of the end product. A recent research project by the usability testing firm Nielsen Norman Group backs up that point. It showed that flat design that eliminates visual cues for links (such as underlining, bold fonts, colored text, etc.) causes frustration and slows users down. People on a site end up guessing where they might be able to click to accomplish something. They hovered over various parts of the page to see whether the elements turned out to be links. This is NOT the user experience you should wish upon your users, so it is not the user experience you should design.
I remember when Apple redesigned iOS and many of the icons for default apps and the design world worked itself up in adoration over the end of skeuomorphism—no more icons and backgrounds trying to mimic real-world objects.
But really, did any non-designer care? Did any regular, everyday iPhone user wake up, turn off the alarm on her phone, and say, “Gosh darn it, that skeuomorphism is driving me bananas!” Did anyone ever have trouble finding the phone app because the icon looked like an ancient corded telephone?
My iPhone was old enough (still is!) that I did not update it to the new iOS, so I never witnessed the new design myself. But I have never heard complaints about the design. The loss of skeuomorphism did not make the software less usable, it seems. But I thought it was odd that the major problem being solved in the design was not usability issues (such as the number of menus one must search through to find various settings) but a problem that only existed to designers.
Flat design is similar. Yes, it looks pretty. And someone who is not a designer can look at a flat design and see a lot of beauty in it. But no matter how many designers are impressed with the design, unless it helps users accomplish their goal, it has missed the mark.
In the case of most websites, you want users to find information or take some kind of action. Hiding what they need, such as masking links so they are hard to notice, does not help.