That...is a pretty interesting combination right there. Hmm, I'd like to see the math over several levels, but it sounds better than I thought. Hmm, a half-orc Fighter/Barbarian Great Weapon Master using Reckless Attack. Interesting!
It's better than you thought only because chocolategravy is exaggerating how many crits you get. From 3rd level to 14th level, a champion gets one extra crit every 20 attacks. From 15th level onwards, he gets one extra crit every 10 attacks. So for 3rd and 4th level, he gets one extra GWM attack every twenty rounds (relative to an Eldritch Knight or Battlemaster), then from 5th to 10th level it's one extra GWM attack every ten rounds (unless he killed or critted something already that round, in which case he would have gotten it anyway even if EK/BM instead of Champion), then from 11th to 14th it's one extra attack every seven rounds, from 15th to 19th it's one every three rounds, and at twentieth level it's one every two and a half rounds.
If you add in a Barbarian 2 multiclass for Reckless attack, you crit 19.5% of the time instead of 5% (27.75% at 17th level when you're Champion 15). At Barb 2/Champion 3 (5th level) you'll be getting 1.19 attacks per round if we neglect the bonus action attack from kills (this skews the analysis slightly in favor of the Barb/Champion) compared to 2.095 attacks per round for a regular Eldritch Knight 5. At Barb 2/Champion 5, the pendulum swings in your favor: you're getting 2.35 attacks per round to his 2.095, and it stays that way up until 11th level when it becomes 2.35 attacks per round to his 3.0975 (again, neglecting bonus attacks from kills). At 13th level it swings back in your favor, at 3.478 attacks per round to his 3.0975, and at 17th level you hit your max attacks/round: 3.623 attacks per round due to superior critical to his 3.0975 attacks per round. At 20th level he pulls ahead again with 4 attacks/round.
Neglecting bonus attacks from kills favors the barb/champ in the analysis, because the EK/Battlemaster benefits more from bonus action kills, since the barb/champ is more likely to have already used up his bonus action on a critical hit.
I think it's clear from the above analysis that GWM barb/champ is primarily attractive for the +10 damage and not for extra attacks (which sometimes don't exist).
The downside to the barb/champ is that 1.) you delay your Extra Attack progression, which also means that your Action Surges are less valuable; 2.) you don't get the EK/Battlemaster special abilities like Precise Strike or Blur. Ergo, it is a pretty good Champion fighter build. It has the virtue of simplicity for players who, like Harry Dresden, deal with a lot of complexity in their regular lives and want just want to play a mindless smashing machine for D&D, and do it more effectively (due to the Reckless plus -5/+10 combo) than a regular Champion fighter.