Flash works on the three major desktop/laptop OS', ChromeOS, all the consoles afaik, and Android. EME will probably result in balkanized DRM that will never work on a bunch of those things (or you'll have to worry about whether the content you want to watch supports your platform).
EME is a step down from Flash for consumer choice. A big fat one.
The Flash browser plugin is NOT available on current versions of Android. There is some Flash available for Adobe AIR apps, but that's something different.
The problem is that copyright holders can still stipulate policies that exclude Linux. For example, http://voddler.com/en/ greets Linux users with "Here you can rent and play movies. For even more movies and TV-series, visit us from your PC or Mac, where we have a even larger selection."
In theory it does. Unfortunately, I've never been able to get it to work because the way in which it obtains a hardware ID to tie playback to doesn't work on the Linux distro I use and I can't feasibly debug or fix it. Assuming it were actually secure as stated, the Mozilla solution would have the same problem - if you can modify how it obtains the hardware ID, you can bypass the DRM in ways that content providers wouldn't accept.
(The really stupid thing is, I was trying to watch online streaming video that was available to the whole of the UK, with access controlled through IP geolocation. There was no need to tie decryption to a particular device ID in the first place. EME seems to be taking exactly the same approach though with support from Mozilla.)
EME is a step down from Flash for consumer choice. A big fat one.