Actually yes, I just verified. Popping berserker rage DOES proc enraged regeneration, I did it tonight in both prot spec and arms spec. You still have to hit its button, but it becomes available, as long as you haven't already used it before its cooldown is up.

Enraged Regeneration requires an enrage effect to trigger/proc/use it, however you want to word it. Same with Raging Blow in the fury tree, and the fury talent Enrage give you a 3/6/9% chance of becoming enraged. Berserker Rage forces the enraged effect.

I think you're remembering how Berserker Rage used to work. If we were cold with zero rage, we would have to use Berserker Rage in order to use any of our shouts. They changed how Battle and Commanding Shout work. Now in addition to their buffs, they generate 20 rage, but have a 1 minute cooldown. Also, BR at one time might have caused us to take extra damage, I don't remember...it doesn't now. Berserker stance also used to make us take more damage...they took that away too, now it's a pure damage boost with no penalty other than the limit on which abilities you can use in that stance.