Your idea doesn't work from a game design perspective.
Players don't like feeling weakened. This is proved through WoW's history. It's better to buff the enemy instead of weakening all player characters. Imagine casting the same spells just to deal less damage for fighting past 15 minutes on a progression boss. People would call bullshit on the nerfs. I could shoot a target dummy or be killing trash mobs for hours, but some raid bosses suddenly nerf me? It makes no logical sense either.
So when presented with this situation, it's just easier to buff the boss and make em go berserk.
Exhaust would make DPS readings incredibly annoying because you'd have to account for it, and then the "You're a caster class Exhaust doesn't effect you as much" and "Screw you DoT whore Exhaust isn't anything to you Exhaust hits MY class hardest".
It'd annoy people, making them go berserk is perfect, you don't stick around, DPS readings don't get boned.
HEH while it is a game and you have to toss logic out the window. There are things in the game that using "WoW logic" don't make sense.
I never wondered that about enrage timers even in the suspended IRL logic world of games I figured it was, as others have said, just getting annoyed at the raid. Like a "ok enough of this crap".
Or you can look at it as the popeye. The enrage is "thats alls I can stands and I can't stands no more" TOOT TOOT...boss eats spinach...
On the flip side while YES I KNOW ITS A GAME...I always questioned things for S&G, like why a boss waits for YOU to pull THEM. "oh hey look a bunch of people just came into my realm/lair and are making a mess of my plans..I should run over there and stop them but they're eating so. I guess it's the polite thing to do."
I think it's logical that after 10 minutes of being zapped and poked with stuff you lose your shit.
Also, there are certain things that make a lot less sense.
For example, as a boss, watching a whole raid of people gathering in front of you, buffing etc. for minutes. Like "Oh, came to slay me? Take your time to prepare!"
Berserk is not necessarily a timer. Sha of Pride HAS a berserk mechanic, it's called Pride. After some time, everyone will have 100 pride, if you don't push the boss into P2 or if you can't kill the boss fast enough in P2 so everyone will me mind-controlled.
That's a soft-enrage, not a "real" enrage..
But it still doesn't change the fact that Heigan have NO KIND of enrage at all...