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Cyber-Dave.
With feats at level 11 at level 11 the two best single class fighter damage builds are polearm master and hand crossbow fighter.
That is true. After that, however, what you wrote isn't how analysis is done. You provided rough guesstimates and left information that hindered your argument out. We need actual targets and actual Armor Classes in order to perform an analysis. You also gave the archer a +2 bonus for a fighting style, but you ignored the fighting style benefit of the polearm user. So, let's try again:
First, we need to create a set of assumptions and explicate them so that our analysis is transparent. Let's assume that the archer, in order to make use of those skill benefits you discussed, is using studded leather (the best light armor). Let's assume that the polearm user is using platemail (the best heavy armor). Let's assume no magic items (as those are a mixed bag and treated very differently from group to group). Likewise, opponents are a mixed bag as well. For the sake of simplicity (as I am not willing to take the time required for a more complete analysis right now), let's pit the two characters against each other. The archer will, as per your assumption, use a hand a crossbow. The polearm user will, as per your assumption, use a glaive. Each will start with the base starting stats assumed by the book. The archer will pick a race that benefits Dex. The polearm user will pick a race that benefits strength. Each will max out each option, but only after choosing the two feats you opted to pick for our example. Each has three feats. That means that each will have a Str/Dex of 19 (not 20) at level 11. The archer will gain a +2 attack bonus for its fighting style. The polearm user will gain a +1 AC. Now let's compare the numbers:
Polearm user: +8 to attack, 19 AC, 3 attacks that deal 1d10+4 (+10 at the cost of -5 to attack), one bonus action attack that deals 1d4+4 (+10 at the cost of -5 to attack).
Archer: +10 to attack, 16 AC, 3 attacks that deal 1d6+4 (+10 at the cost of -5 to attack), one bonus action attack that deals 1d6+4 (+10 at the cost of -5 to attack).
The average damage of the polearm user against the archer without taking a -5 to attack is: (0.6*9.5)+(0.05*15)=6.45 and (0.6*6.5)+(0.05*9)=4.35 for a total of (3*6.45)+4.35=23.7
The same numbers for the archer look like this: (0.55*7.5)+(0.05*11)=4.675 for a total of 4.675*4=18.75
If both start using their +10 damage/-5 to attack, the numbers then look like this:
The average damage for the polearm user is: (0.35*19.5)+(0.05*25)=8.075 and (0.35*16.5)+(0.05*19)=6.725 for a total of (3*8.075)+6.725=30.95
The average damage of the archer is: (0.3*17.5)+(0.05*21)=6.3 for a total of 6.3*4=25.2
In other words, in melee, the polearm wielder does 5.75 more points of damage per turn than the archer. The polearm wielder also threatens, which the archer does not. In return, the archer will have range and between +5 and +1 higher initiative skill check bonus with a few (arguably) more effective skills than the polearm wielder (depending on build).
Taking this white room set of assumptions, and using your "4 more potential attacks per combat," scenario, the polearm wielder will be identically effective to the archer any time combat lasts more than 4.3826086956521739130434782608696 turns. Most combats, in my experience, last around 4 or 5 turns. As each of these two combatants will have upwards of 65 hit points, and possibly as much as 109 hit points, we are looking at a fight that will last at least 3 average turns (if each build is gimped in the Con department) and possibly as long as 4.3253968253968253968253968253968 turns worth of attacks for the archer to kill the polearm user and 3.5218093699515347334410339256866 turns for the polearm user to kill the archer. So, if the fighter is an idiot and doesn't attack for one turn (because the archer is between 35 and 60 feet away), the archer has a 0.19641254455470933661563710028979 turn advantage on the fighter (assuming optimized Con, as that seems like the logical choice for both builds). If the fighter plays it smart, numerically speaking, he will get one round of javelin attacks in before closing. (I'm getting lazy, so I won't bother calculating, but I think the average damage off of 3 javelin attacks--made with disadvantage--should offset that 0.2--rounded off--turn advantage). Hell, if the polearm user has a dex of 12 or 14, it might opt for a round of longbow attacks instead (which will definitely set off that 0.2 advantage). On the other hand, if they start in 30 feet or less range, no matter how smart the archer is, the polearm user has a 0.8035874554452906633843628997102 turn advantage.
As I said, in my opinion, that is balanced. Especially because this is a white room assessment. In reality, things play out very different. Often, there is terrain that allows targets to take full cover (no attack possible) against ranged attackers. Often, combats are set up in ways that force melee range combat. And often, characters get to start at even greater range, and the ranged characters dominate (though in my experience the fighter's don't sit around doing nothing; instead, they pull out bows or javelins, and they act as backup). It's up to the DM to make sure that all characters get their spotlight moments by designing his campaign/encounters appropriately. The white room, however, does not suggest what you have claimed it does.
Rather, according to the whiteroom, the melee fighter has a drastic advantage if the fight starts in 30 foot range, has moderate disadvantage if the fight starts between 35-60 feet and he is an idiot, is evenly matched if the fight starts between 35-60 feet and he remembers that he knows how to use more weapons than those he has specialized in, and has a drastic disadvantage if the fight starts at 65+ foot range.