I like the idea and most of the execution. Maybe allow for the fury die to be applied to grapple checks as well, to help give some utility?

Name suggestion for Poor-Man's Smite: Shattering Blow. Further, "your fists count as separate weapons for the purposes of two-weapon fighting," doesn't grant the light property in 5e.

Two-Step Goodbye should probably apply specifically it damages the creature hit. Just to establish what happens when it affects a creature to become unstable when reduced to 0 hit points instead of dead when reduced to 0 hit points.

I'm not clear how Overkill interacts with Brutal Critical and a critical hit. I'm assuming the extra dice only apply to the initial fury die, for an average of 6.5*3+5.5+4.5+3.5+2.5+5+3=43.5 average damage on a hit using a d12 fury die at 14th level. In that case, it may be easier to write it as "when you score a critical hit, you can deal additional die of damage of each die size smaller than your current fury die," or something similar. Although I'd be interested in how you got to "~60 damage," for this.

However, I read the idea in general as too strong. Contrast a rogue at 14th level, who deals 7d6+1d8+5 or 34 average in the same circumstances. Now, the rogue can apply crit damage to that, for 63 damage, but the rogue can't do that at will; they have to actually score a critical hit. Can't do this every turn, but it still steps well into the rogue's features. The barbarian class is already just below the rogue's damage per turn anyway, and the class has far more bulk to boot.