Only in theorycrafting does a class with two d10 attacks and one d4 attack do more damage than a class with four d12 attacks~
At first I was going to call this a bad faith argument, but on further reflection I realize that not everyone is good at math, and, while I've posted a link to my spreadsheet for others to examine, not everyone is going to be willing to do that, so that's not the best way of explaining things either. My charts can thus come off as being supported by "Trust me, bro".
So to clear things up, here's a manual breakdown on the math involved.
You seem to be alluding to a barbarian using PAM vs a level 17+ monk, so I'll set things at level 17. Max attribute is 20, and both classes will be assumed to have reached that by that level.
The dice averages for their attacks are 5.5 + 5.5 + 2.5 = 13.5 for the barbarian and 6.5 + 6.5 + 6.5 + 6.5 = 26 for the monk. That clearly favors the monk.
Then you add the attribute mod — +5, 3 times for the barbarian and 4 times for the monk. Now we're at 28.5 vs 46. Favors the monk even more.
Then you add Rage damage for the barbarian — +4, 3 times, putting it at 40.5 vs the 46 of the monk. Significantly closes the gap, but still favors the monk.
Then you add the Graze weapon mastery. This requires factoring in accuracy. For an accuracy between 60% and 80% (a reasonable range to examine), the barbarian's total (with Graze) becomes between 30.3 and 35.4, while the monk has between 27.6 and 36.8. There's complete overlap now, with the monk potentially doing either less or more than the barbarian. They're almost tied at 70%.
Note: The monk isn't getting any weapon mastery because he's using unarmed strikes (1d12), and there are no weapon masteries for unarmed strikes (yet).
Calculation for 60%: (5.5+5+4)*60% + (5*40%) = 10.7; (2.5+5+4)*60% + (5*40%) = 8.9; 10.7+10.7+8.9 = 30.3
Calculation for 80%: (5.5+5+4)*80% + (5*20%) = 12.6; (2.5+5+4)*80% + (5*20%) = 10.2; 12.6+12.6+10.2 = 35.4
Then you add Great Weapon Master damage. This is useful for the barbarian, but not for the monk (H2H does not qualify for either of the GWM features). Monk doesn't have an equivalent feat for any additional damage boost (aside from Charger, which both can get). That's +6 damage once per turn, between 93.6% and 99.2% of the time (we're not using Reckless Attack yet, but its addition would be just a fraction of a point for this). +5.6 to +5.95 pushes the barbarian to between 35.9 and 41.35 vs the monk's 27.6 to 36.8. And now the barbarian is ahead. (+12% to +30%)
While the barbarian can use Reckless Attack (and gain more damage from Frenzy if it's the Berserker subclass) for advantage, the monk can use Stunning Strike for advantage as well, so I'm not going to do the math on that. The gains will be similar on both sides, though the barbarian's will be more reliable.
I didn't include Charger. Both can gain the same benefit from it, though monk is probably more likely to take it as a feat, in which case it mostly balances out the benefit of GWM, and the two classes are again closer to tied.
So, discounting advantage from Reckless Attack/Stunning Strike, not counting bonus feat damage from GWM/Charger, and not including subclass damage features, the monk and the barbarian generate similar damage output. It does require weapon mastery on the barbarian's part, but also that the monk always uses Flurry of Blows. A more realistic FoB usage rate would largely balance out the lack of weapon masteries, so the PHB damage is likely also similar.
But yes, a huge part of Monk Bad rhetoric is the idea that Monks should be able to outperform every other martial in every regard, and if there is a single area that Monk is not consistently superior in, they are useless as a class. They want the class to be able to do more damage than every other martial, move three times farther than anyone else, have superior defenses in every regard, and throw out multiple crippling conditions per turn—and they think the Monk should be able to do all of these things simultaneously and for either inconsequential or even nonexistent resource cost.
I disagree with the premise. While the monk and barbarian are comparable at baseline, the barbarian has a much higher potential. The ease of using Reckless Attack, not needing to sacrifice your bonus action to use other class features, and (in the PHB) the power attack from GWM vs low AC enemies gives rise to massive potential damage. Alternatively, the UA Berserker's Frenzy plus the new GWM can do similarly. Those damage spikes are memorable, whereas the normal damage isn't. (See also: paladins)
Basically, the monk does OK damage, but it
only does "OK" damage. It also does lots of other things (which I think is a large part of the draw of the class), but those other things being so heavily limited by ki/discipline points means that players get neither amazing damage nor amazing thematics (until later levels).
The suggestions I have seen tend to try to boost one or the other. Maybe give Flurry of Blows a third attack in order to be a greater threat on the battlefield, or provide ways of using "lesser" versions of class features without spending DP, so you can still always do monk-like things.
And, for me at least, comparisons with the barbarian (or any other class) is purely as a benchmark to make sure suggested changes do not reach excessive levels. I've spent a lot of time on the barbarian and fighter since UA5, and since they largely feel pretty balanced for their intended purpose, I feel they make good references when examining how other classes behave.
