I am going to comment as I read thru, and then pull it together at the end. I specifically did not read anyone else's comments so that I come at it without preconceived notions. If I don't mention something someone else said don't take that as disagreement with them, but rather that they might be more observant than I was.
First point - your subject says Fighter archetype, may want to change that.
Good ribbon, though since you need to either attack or take damage to keep raging, you rarely will have you action available to make checks. Still, sets the scene well.
Renewable resource, but one you don't start any combat with so you can't nova early on. And only available in combats you rage -- nothing during other combats. That balance works for me.
I think there are too many options for quick play. They're cool, I wouldn't suggest trimming. Perhaps pick 3 to start and gain another at two other levels. Oh, and the wording could be read that if you crit you get 3 momentum - one for a successful melee attack, and two for rolling a crit. I'd touch it up slightly to make sure it's what you want. It is also unclear if you gain Momentum if you crit when taking an attack on someone else's turn. Another small benefit of starting with fewer is make it less powerful for cherrypicking with multiclassing.
I like them all, want to talk about two.
Brutality I worried if it wasn't giving you enough, but rereading it to see it's chosen after the hit like a paladin's smite puts it back into perspective.
Fearsome is perhaps the most powerful of them with both a great effect and a good ability of a successful save - but the DC is off CHR which is not usually an oft-raised Barbarian ability. I'll be watching while I read if CHR is a focus.
Cool power, not sure if it's enough for a level. If you do go for a limited selection of Momentum powers, this would be a good level to get another. If not, some out-of-rage/out-of-combat ability that helps reinforce the descriptive theme. (Or both.)
Cool. My only comment is that at 10th, the other barbarian subclasses give abilities that can be used outside rage. This isn't a problem, just a comment. At this point, with 4 rages a day, you are likely raging in the majority of encounters anyway.
I admire your restraint - often when PEACH I see either features that are too powerful by themselves, or are all multipliers of each other. This one, in having to give up the bonus of Extra Attack as well as only usable after the combat has been going (to get 3 Momentum), and a debuff that gives saves every round (a good balance for no Concentration). Throat Punch seems really good, and many others can pare well with group dynamics, but as a 14th level ability I would expect it to have meat to it.
Conclusion
From a balance perspective I think you did a good job, maybe a few places I'd tweak the power level up as mentioned above. Oh, and there isn't the CHR focus elsewhere that would make me concerned about Fearless being too good, so that's still on point.
On thinking back, I think I'd like to add another ability at 14th - when you rage, you gain one momentum die. Just to get things moving at that high level.
I am a little concerned that there's no abilities that work outside rage, and to a lesser degree (based on other barb subclasses) no abilities that work outside combat.
From a flavor perspective, there are only a few features (Bestial Nature, and Fearsome) that make this subclass a better fit for the description then one of the other subclasses. You may want to add in some additional ribbon, or substitute different features that really bring the more-beast-than-man concept home. Some random ideas:
1. Gain advantage on CHR skills vs. beasts. (So when calm can be one with them, and at all times can intimidate and freak them out). Maybe gain Survival skill so they can live as beasts. (Add to Bestial Nature?).
2. Increase Momentum die by 1 size when using Unarmored Defense. (Gained early to give incentive to play to type)
3. Natural weapons while raging? Eh, that would likely not be better than actual weapons, so maybe just put them at the same level of good pugilists and give them d4 for unarmed damage at all times, and they can use it with Reckless Attack.
Just a few ideas off the top of my head, don't take that as exhaustive.