Okay, so, let's start from your wrong premises
Max Damage does not matter - average does
And you are using the wrong weapon for GWF based on that - Greatsword is better than Greataxe. Greatsword is already superior to Greataxe, but it becomes even better with GWF, when the probability of rolling 1s and 2s is higher.
For reference, the average damage from 2d6 with GWF is \$\approx 8.33\$ (you reroll 1s and 2s). The average damage of 1d8 is \$= 4.5\$.
So, for TWF, you deal damage equal1 to $$d_T = (a + 1) \cdot (4.5 + \textrm{Str})$$
where \$a\$ is the number of attacks you have and \$\textrm{Str}\$ is the STR Modifier. For GWF, you deal $$d_G = a \cdot (8.33 + \textrm{Str})$$
We can check when things start going badly for TWF, e.g., for a +5 STR modifier (which, by the way, is not exactly a common roll for a 5th level character, but let's bear with your premise)
$$13.33a > 9.5a + 9.5$$
$$3.83a > 9.5$$
$$a > 2.48$$
So, at 5th level (\$a = 2\$), you are still doing more damage with TWF than GWF. After 11th level, with one more attack, that's no longer true. It gets worse and worse.
Action-Economy
Another thing is: for that extra attack you are using your bonus action. Also note that features like Action Surge will benefit GWF more, since it essentially means \$\tilde{a} = 2a\$, not allowing you to get an extra bonus action. So, for example, in your Barbarian Multiclass, you would be unable to Rage and make the extra attack in the same turn. As the levels go by, you also get more and more uses for your Bonus Action that will not be used because you are using it for an extra attack.
Opportunity Attack
If you are able to make an OA, GWF is superior again, since it's equivalent to increasing your \$a\$ by 1. There isn't much to say here other than that.
Better damage dealing feats
Great Weapon Master exists. And it is insanely good. And I'm not even talking about the extra attack you get on critical or killing hits, I'm talking about the second bullet.
Before you make a melee attack with a heavy weapon that you are proficient with, you can choose to take a -5 penalty to the attack roll. If the attack hits, you add +10 to the attack's damage.
This increases your DPR considerably. On the other hand, TWFs have... Dual Wielder.. which is not as amazing2, sorry.
Stacking magic weapons
I'm not sure what you meant by Stacking magic weapons, but if you have two +3 weapons, they don't give you a +6 bonus.
You have a bonus to attack and damage rolls made with this magic weapon. The bonus is determined by the weapon's rarity.
This means it's actually harder to get magic items as a TWF, because if you want to apply the +3 in every attack, you need TWO +3 magic weapons, while GWF only needs one.
1 Note: the actual damage would be \$p \cdot d_T\$ and \$p \cdot d_G\$, where \$p\$ is the probability of actually hitting. I'm assuming both have the same attack hit modifier (i.e., +5), so they have the same probability of hitting, thus the probability is irrelevant here3. It changes considerably the comparison of DW vs GWM though, thus the math is left aside for that one. If you want the math, I suggest asking a new question about it.
2 Okay, I got a little confused here - I went by OP's words on this one, but then I double checked and Dual Wielder does not give you a reaction or +1 hit/+1 damage. Either way, GWM is usually better than a +2 ASI, which would give you exactly the +1 hit/+1 damage.
After the edit, OP mentions Blade Mastery, but that can also be used with Greatsword, so all it does is increasing \$p\$ for both the styles (see below). As V2Blast mentioned in a comment, Blade Mastery doesn't give you +1 damage either, but let that aside.
3 As noted by nitsua the cool sheep, it's not completely irrelevant because critical hits exist. I'll make the actual math here and let the body as it is. Let \$p\$ be the probability of a normal hit, given by
$$p = (21 - \textrm{AC} + B) \cdot 0.05$$
where AC is the AC and \$B\$ is the modifier to hit, and \$p\$ is capped at 0 and 0.95 (i.e. you always miss rolling 1 and the probability can't be lower than 0). Let \$d_{Ta}\$ be the average damage per attack that hit for TWF, \$d_{Tc}\$ be the average damage per critical hit for TWF, \$d_{Ga}\$ be the average damage per attack that hit for GWF and \$d_{Gc}\$ the average damage per critical hit for GWF. The exact average damage of TWF is given by
$$d_T = (a + 1) \cdot (p \cdot d_{Ta} + 0.05 d_{Tc})$$
and the average damage of GWF is given by
$$d_G = a \cdot (p \cdot d_{Ga} + 0.05 d_{Gc})$$
Choosing your values of STR modifier, we have \$d_{Ta} = 9.5\$, \$d{Tc} = 13\$, \$d_{Ga} = 13.33\$, \$d_{Gc} = 21.66\$. Assuming 14 AC (quite usual), we have \$p = 0.6\$, giving us
$$d_T = (a + 1) \cdot 6.35$$
and
$$d_G = a \cdot 9.081$$
Finding \$a\$ again, we have
$$9.081a > 6.35a + 6.35$$
$$2.731a > 6.35$$
$$a > 2.325$$
So, taking into account critical hits actually makes GWF become even better - because the damage on the dice rolls is larger for GWF.
Bonus Action Limitations of Two Weapon Fighting
Raging is a bonus action. If you want to rage, you'll be unable use two weapon fighting that turn. This can put you at a disadvantage right from the start of combat.
Great Weapon Master Bonus Actions
It's not just a critical attack that allows a great weapon master to attack with their bonus action.
On your turn, when you score a critical hit with a melee weapon or reduce a creature to 0 hit points with one, you can make one melee weapon attack as a bonus action.
This is still conditional, and will depend on the kind of opponents the DM throws at the party, but mobs of foes and minion types often make appearances in many campaigns and will greatly boost the damage of a great weapon master.
Growth from Extra Attack is in favor of great weapons
With a great weapon, the Extra Attack feature makes you go from 1 attack per turn to 2 (barring great weapon master conditional bonus action attacks). With a two weapon fighter, you go from 2 attacks to 3. So while the damage of the great weapon increases by 100%, the two weapon fighter only gets a bit more then a 50% damage boost (unless you have the two weapon fighting style, in which case, it is a 50% increase).
This same issue also means a great weapon barbarian benefits more from the single extra attack granted by the spell Haste. It also pops up if you decide to multiclass into fighter, because action surge doesn't grant you another bonus action.
Problems created from using more than one weapon
There are many spells you could potentially have applied to your weapon by a party member, such as Magic Weapon
2nd Level Transmutation Spell
You touch a nonmagical weapon. Until the spell ends, that weapon becomes a magic weapon with a +1 bonus to attack rolls and damage rolls.
At Higher Levels. When you cast this spell using a spell slot of 4th level or higher, the bonus increases to +2. When you use a spell slot of 6th level or higher, the bonus increases to +3.
However, this spell and many others only work on a single weapon at a time. So while a great weapon user has all their damage become magical (allowing it to bypass resistance to non-magical damage, which is critical against many foes), a two weapon fighter is left with only part of their attacks improved.
Consider your role in the party
Reckless attack isn't just a risky move, but a way to control opponents. Granting your opponents advantage encourages them to attack you, rather than your squishy wizard friend. With rage, and especially bear totem rage, you can take the damage. If your build is too defensive and never uses reckless attack, opponents may ignore you and go after bigger threats rather then one they can barely hit and can't deal full damage to.
Advantage has a huge effect
This is kind of obvious, but maybe it'll help illustrate just how useful reckless attack is for someone with great weapon master.
Source
If you look at this chart, you can see you have an 75.1% chance of rolling a 11 or higher on your roll if you have advantage. With 18 strength and a +2 proficiency (what you'd probably have at level 4), that's enough to hit 12 AC target, which is the AC of plenty of big beasts and smaller enemies. You'll still hit 15 AC targets 57.8% of the time (without advantage and without the -5 to hit, you'd normally hit 60% of the time with the same stats).
And against very high AC targets, you can choose to not use the power attack and still be relatively effective.
Best Answer
Berserker is vulnerable to losing their rage
From the description of the rage feature:
This means that if you don't have a target within javelin range, possibly due to full cover, and haven't taken any damage, your rage ends at the end of your turn. Your DM might rule that you can hit yourself, but that strongly depends on your DM. It also means a disabling status effect, such as that inflicted by hold person or sleep, can mean your rage ends unless you get a party member to help you.
Even if you simply use rage again, it costs a bonus action, so you cannot make a second attack on that turn, and you are also exhausted, meaning you have a disadvantage on ability checks, meaning it's harder to avoid grapples/shoves (though rage's advantage on strength checks means you'll be rolling normally, at least).