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.