The title of that email is a bit dramatic.
It's been known since at least last year that basically all browsers are removing Plugin support for security reasons. Unity's Web Player is a plugin, ergo it won't work.
WebGL is the new solution, using browser's built-in asm.js and some magic with IL2CPP you can deploy directly to WebGL and it is natively supported in modern browsers.
WebGL currently is a bit clunky, but it is getting better. I've done some builds for clients and it is a bit hit-and-miss depending on your settings, their internet speed, scene complexity, build size, browser type and version. Eventually this will all catch up and stabilize. You need to use the NACL version of Playmaker (included in the download as a separate .unitypackage) which strips networking for it to work on WebGL.
IMO, Kongregate and similar websites should have jumped on this a long time ago.