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The thing is the MMA fighter practices enough boxing to be aware of what the boxer is going to do, whereas the boxer probably doesn't practice any take downs or ground fighting unless they do it as a hobby. Those take downs are going to hurt a lot more in the street too, and as others have mentioned, they're in a better position to fight dirty than the other guy.
"Getting a takedown doesn't stop someone from grabbing your balls and squeezing or slamming your head on concrete"
But the MMA fighter would be better prepared to fight on the ground, including doing those things, breaking your limbs, etc >

And further the whole mythology the martial arts and fighting sport communities have built around "street fights" is just silly and 99% of it is stupid asshole macho chest puffing.
Here's the truth.  And it's the truth that any self defense instructor who actually cares about you will tell you.
In a fight the bigger person is usually going to win.  If the bigger person gets in a good solid first strike, he's almost always going to win.  If the bigger person manages to get in a good solid first strike when you are not ready for it, taking you by surprise, where he is already mentally "on" and ready for a fight and you are not, he's always winning within a statistical margin of error.  All the elite black belt ninja navy seal tricks don't matter because you've already lost the fight before you're in a headspace to deploy them.
Self defense is not about fighting.  A good self defense strategy is a handful of solid techniques and strategies to survive the opening moments of being taken by surprise so you can counter just enough to get fuck out of there. >

This is bs. There’s almost always a bigger person in a fight. Being bigger doesn’t mean being physically or mentally tougher than the other person >

If you take the fight out of boxing rules the MMA fighter will win almost every time. Boxers train to fight within boxing rules which are very strict. 
Hell if you put the fight in boxing rules but with MMA gloves I think most MMA fighters would win. >

I remember a project a few Gracie black belts did where they analyzed over 100 hours of street fight footage. Whoever got the first takedown won over 90% of the time, and over 90% of the KOs from stand up came from the first punch. 
So it’s a bit of a toss up, if the grappler initiates grappling first, they win, if a striker strikes first, they win.
The issue with ball grabbing and eye gouging a grappler on the ground is that they’ll do the same to you while being significantly better on the ground. Also a good grappler immediately transitions from a takedown to a sub or dominate position against a non grappler.
The biggest problem with grappling as self defense is if your attacker has friends, theyre gonna stomp you to death >

Here's the thing.
In MMA there's absolutely nothing stopping you from being a boxer.  You can step into an MMA ring and just box, literally nothing is stopping you.  You won't be breaking any rules. 
There's reason nobody does that. >

I mean this isn't an unpopular opinion in the way framed your opinion. In the street fight as you presented they arent conforming to either MMA or Boxing rules where the goal is to just knockout or cause your opponent to concede. If you want your opponent dead neither is ideal. It would come down to individual physique and pain tolerance. >

Need to define street fights too lol.  Are we including knives, clubs, barstools, brass knuckles? >

First to grab the balls or the eyes win. >

Boxers actually do very well in street fight compare to other combat sports. 
Also getting hit the the ball isnt gonna stop a fight >

I think judo and BJJ would be the difference. If the boxer cannot get in range (kicks) he will not be effective. If they do manage to get close enough, you could throw them using the judo.