OK - Seems to me the balance here is about right between Joe McPolearm, and Susan McCrossbow. If you check the numbers using a non-trivial to-hit chance (say 1d20 + 10 Vs AC 16 = 75%), the damage output in pure numbers is basically the same with a nudge to the Hand Crossbow
Joe has GWF style, GWM, and Polearm Mastery. Susan has Archery Style, SS, and CE. They are both level twelfth level and so get 3 attacks
Joe:
To hit chance with GWM = 50%
Average Damage per attack = 6.3 (1d10 rerolling 1&2) + 5 (STR) + 10 (GWM)
Average Damage with Polearm bonus = 3 (1d4 rerolling 1&2) + 5 (STR) + 10 (GWM)
Total = 81.9
Total with Hit % = 40.95
Susan:
To hit chance with GWM = 60% (+10% from Archery Style)
Average Damage per attack = 3.5 + 5 (DEX) + 10 (SS)
Average Damage with CE = 3.5 + 5 (DEX) + 10 (SS)
Total = 74.0
Total with Hit % = 44.4
So conclusively, Susan does more damage per turn against Combat Dummies to the tune of about 1d6 per round. Case closed.
Now there's another argument about AC here, in that all things being equal, Joe has an AC 1 higher than Susan at level 12. In context this is a 5% damage reduction. Against a CR12 Erinyes which can do 60 points of damage per turn on average on 3 hits, and factoring the to-hit % benefit from an AC of 1, Joe takes 3 points of damage less per turn on average, or about 1d6 per round. Therefore, you are transferring damage output, to damage mitigation. This is a valid choice in design between Joe and Susan (Yes Susan can wear Plate but at Level 12 she wont have the ASI necessary for STR 15 unless CON is dropped, which basically mitigates the benefit from AC anyway)
Now, here's where things get interesting.....
On an infinitely flat plane, Susan is now utterly badass. With her 120ft range, she's dropping every melee enemy before it gets a hit on her (assuming she has the ammo) by moving and firing each turn. Joe by contrast wastes rounds running up to it. Flying creatures give Joe a headache. Not so Susan (assuming the don't fly more than 120ft away).
However, Move indoors and things change. Most rooms are less than 30 feet to a side, and so most enemies will reach everyone inside in 1 round. Joe now doesn't have to worry about wasting turns - enemies come to him. Susan's higher Initiative helps attack first, but that means moving and holding an action every turn to attack something running at her (Again assuming she isn't jumped by something she didn't see - Fighter's tend towards low perception/investigation). In this Joe begins to shine because of a better AC (5% damage mitigation), GWM, and Opportunity attacks. And for Joe that OA range is 10ft, not 5. Immense!
Once Joe has something pinned down, they can't leave or they take an attack, increasing his potential DPR. This is supplemented by GWM in confined spaces - when he kills something he gets a free attack. When his Crits (5% of the Time) he gets a free attack. This constantly ups Joe's DPR when in melee, preferably against 1 or more enemies. Joe loves it.
By contrast something can run at Susan, attack, then dart behind a door (or behind a tree in a densely packed forest). Susan doesn't want to be in melee, but she can't open a door without being 5 feet. She can't stop them doing it because she hasn't got a melee weapon, and even if she has her DPR bonus from it is going to be poor compared to Joe - all her perks are focused on ranged weapons. It potentially gets worse for Susan if things are tight AND she's got allies - LoS becomes a bit of an issue, and she's relegated to just dealing damage. By contrast, Joe is locking down enemies, protecting the group, and dealing possibly more damage that Susan. In a tightly packed space, frankly, Joe is a machine.
So what does that tell us purely from a DPR standpoint. In a situation that favours a Ranged attacker, Susan McCrossbow is better. In a situation that favours a Melee Attacker, Joe McPolearm reigns supreme.
So a Ranged based Fighter is better at Ranged, and a Melee based Fighter is better in Melee.......huh, Who'd've Thunk it.....