He has a point, for example now they've finally stopped supporting IE11 you can legitimately say to customers that supporting IE11 is no longer possible because it's a security hole.

Extended support by MS for Windows 7 means the only reason you're not supporting it yourself is your lack of motivation, and so to your customers (and your own upper management), for a big contract that's not going to wash.

You might even find that their contract even says something to that effect.