It's legit, complicated, and about as powerful as it should be
A monk that takes the archetype master of many styles can take at level 1 as its monk bonus feat the feat Pummeling Style, take at level 2 as its monk bonus feat Pummeling Charge, take at level 5 the feat Martial Training I and at level 7 the feat Martial Training II. Such a monk at level 7 can charge and, at the charge's end, unleash his full flurry of blows of unarmed attacks—that's 3 (4 is but a level away) that each deal 1d8 + Strength modifier points of damage and an extra +1d6 points of damage—on a lone foe. This is acceptable but by no means remarkable, the archetype mandates playing a (not unchained) monk, and such a character might be better served by multiclassing into the base class brawler.
Speaking of brawler, a creature can instead take at level 1 the base class brawler (which has its own unarmed strike damage progression), at level 3 take the feat Martial Training I, at level 5 take the feat Martial Training II (picking the Broken Blade discipline and this second feat allowing the brawler to take the stance pugilist stance which the GM may rule scales as the table describes rather than not scale as per its description), at level 6 employ the brawler class feature martial flexibility to gain temporarily the feat Pummeling Style until at level 7 that feat can be taken permanently, and—finally!—at level 12 again employ the brawler class feature martial flexibility to gain temporarily the feat Pummeling Charge until at level 13 that feat can be taken permanently.
Then, at level 13, the brawler can charge and, at the charge's end, unleash his full brawler's flurry of unarmed attacks—that's 3 from the brawler's base attack bonus and another 2 from the virtual Two-weapon Fighting and Improved Two-weapon Fighting feats that all deal 2d6 points of damage and an extra +2d6 points of damage—against a lone foe. Assuming a decent Strength score, the feat Power Attack, and maybe a magical buff or two, this is pretty okay and about where a creature of that experience level should be.
(Keep in mind that pugilist stance deals extra dice of damage, and that extra dice of damage, for example, are not multiplied on a critical hit (see Combat on Multiplying Damage). Thus a level 13 brawler like the one above on a typical hit deals with his unarmed strike 2d6 + his Strength modifier +1d6 or 2d6 points of damage depending on the GM rules on Pugilist Stance, but on a typical critical hit with his unarmed strike he deals 4d6 + twice his Strength modifier +1d6 or 2d6 points of damage, not 6d6 or 8d6 + twice his Strength modifier!)