Flat design is great but it can be even better if used properly

As usual, the NN/g blog has rich content about how to improve your customer's user experience.

I quote:

Flat design, especially when combined with strategic minimalism, can be a powerful aesthetic tool. It can convey a sense of luxury or trendiness, and in some cases can be used to appeal to young adult users.
Unfortunately, flat design has a major flaw: it often leads to click uncertainty and decreased user efficiency. When designers flatten the UI, they tend to remove many signifiers that normally tell users where to click. 

Our experience of helping companies improve their customer touch points has found this to be a big, big problem. The web site looks really good, but older and less experienced users are left to try and navigate between the pretty imagery and the navigation links

Unlike most consultancies, NN/g not only identifies the problem it tells you what to do about it. I strongly suggest that you read it and decide if your web site makes these mistakes.

You will then have to persuade your designers that the usability of the site is more important than its asthetic  look. Good luck. Dick Stroud