You're looking at it from the wrong perspective: over the long term it increases your average damage because you hit more often and therefore do damage more times than you would if you were trying to hit without Advantage.
Okay, but again, making
two attack rolls is always better than making
one attack roll with advantage. Like, we agree that this is mathematically correct, yes?
A single normal attack roll has X% chance of success to deal N base damage. X will vary based on the stats of the attacker and the target. N mostly depends on the attacker's stats, but may be affected by something like Resistance. Since accuracy is something you want factored in, we have to account for approximate hit rate, which (based on much more in-depth number crunching done by others) is generally in the 60%-65% range, varying up or down only about ten percentage points in either direction in most cases. And since we're talking about Barbarians and Champions--two things that love crits--we're going to be looking at things that want big singular damage dice (due to dumb rules regarding 2d6 weapons vs 1d12 weapons).
Given the above, I hope you agree it is a reasonable assumption that we're looking at a 1d12 weapon such as a greataxe, wielded by a character which has reasonably high (16 initial, later maxed) base Strength, no early feats (none before 12th level anyway), and is generally race-agnostic (e.g. I'm not going to make special calculations for Half-Orc Barbarian vs Halfling Champion--if both the Champ and the Barb are the same race, there should be no net effect).
Since the Berserker is generally agreed to be Not Very Good, we'll assume a Totem Warrior. Only one of the Totem Warrior features directly affects damage or accuracy: the 14-level Wolf and Elk totems permit knocking things prone, and the Elk totem permits some extra damage if a save is failed. However, since I presume we are
already assuming Barbarians use Reckless Attack 24/7, this won't actually affect the calculations. To start us off, I'll do calculations at 5th and 11th level--the points where new damage features come online. I will assume no magic weapons for these characters, though if we go to high level I will assume a +2 weapon. (It's worth noting, not having a magic weapon is very favorable to the Barbarian, as increased base damage and increased base accuracy are both more useful for "two ordinary attacks" than they are for "one attack with Advantage.")
At 5th level: Champion has one 3-round use of Action Surge per short rest. Barbarian has 3 rages per day. Assuming the expected "about two combats per rest" pattern (which is in keeping with the 6-8
medium encounters guideline), each character gets ~1 combat where their special bonus doesn't apply, and ~1 combat where it does (Barb may get unlucky, but we'll assume the pattern holds). Obviously, if we tilt things so far that there are only 2 combats a day and either one SR or none then things get wonky, but at that point classes like Paladin and full-casters have such an overwhelming advantage it's not really worth doing the calculations to begin with.
Champion's damage per hit: 1d12+3 = 10.33 on average (due to GWF)
Champion's damage per crit: 2d12+3 = 17.33 on average
Barbarian's damage per hit: 1d12+3 = 9.5 on average, 11.5 when raging
Barbarian's damage per crit: 2d12+3 = 16 on average, 18 when raging
With a 65% chance to hit, the Fighter hits 55% of the time and crits 10% of the time. The Barbarian, if always Reckless, hits 78% of the time and crits 9.75% of the time. If we assume a four-round combat, so the Barbarian
does get to benefit from at least 1 round of Rage that the Champ isn't Action Surging, the Champion makes a total of 14 attack rolls, while the Barbarian makes a total of 8 attack rolls with Advantage.
Champ's average damage (w/3 rounds AS): 14*(.55*10.33+.1*17.33) = 103.803
Barbarian's average damage (w/4 rounds Rage): 8*(.78*11.5+.0975*18) = 85.8
Champion is ahead. But perhaps that second fight, without Rage but with Reckless Attack, will make up the difference?
Champ's avg dmg (4 rounds, non-AS): 8*(.55*10.33+.1*17.33) =59.316
Barb's avg dmg (4 rounds, non-Rage): 8*(.78*9.5+.0975*16) = 71.76
Turns out...not quite? The gap between the first two is 18.003 (exactly), and between the second two is 12.444 (exactly). That's about a difference of one attack. So the Champion has definitely pulled ahead in this pair of combats.
I'll do more math work later if you're interested (the 11th level stuff is more complicated because
number of crit dice changes), but...I'm pretty sure the gap will just get bigger as Proficiency bonus goes up. Again, just for simplicity,
two actual attacks is almost always better than
one attack with advantage, because of static damage modifiers (Strength score, in this case, and a possible contribution from a magic +N weapon).
