I mean, you had the right idea. If you want to do MASSIVE damage on a crit, then a half-orc/barbarian with a greataxe is your best bet (Do Savage Attack and Brutal Critical stack??? 4d12 damage!!?!)

Doing completely unnecessary math, if you have Savage Attack OR Brutal Crit, then a greataxe will do more expected damage if you hit on a 15+ due to the increased crit damage. Otherwise, a greatsword is better

If you have both (or Brutal Crit at level 13) for 2 additional weapon dice, then greataxe is better than greatsword if you hit on a 9+

If you can add 3 additional weapon dice, then greataxe is better than a greatsword if you hit on a 3+

If you can add 4 additional weapon dice, then greataxe is strictly superior.

Maths

Damage formula is (normal damage * number of times you'll hit for normal damage + crit damage ) divided by 20

Greataxe average damage over all rolls on the d20 if you need a roll of X to hit = [1d12*(20-X)+(2d12+1d12*N)*1]/20 where N is the additional number of damage dice

Greatsword avera damage over all rolls on the d20 if you need a roll of X to hit = [2d6*(20-X)+(4d6+1d6*N)*1]/20 where N is the additional number of damage dice

Solve for X such that greataxe damage > greatsword damage