You've misunderstood the ruling.
Addressing a nuance in the PH errata: the rule lets melee weapon attacks use unarmed strikes, despite those strikes not being weapons.
In case you need further proof, you can scroll a bit down the page you linked to:
Unarmed strikes never should have appeared as weapons, hence the correction. The monk is barely affected.
And even further down, we have this:
The unarmed strike mistake caused a number of problems in the system. See Dueling in the fighter for one example.
So, in other words, not only are unarmed strikes no longer weapons, they were specifically changed to prevent exactly what you're trying to do.
Now, if we want to ignore Crawford's Twitter and just look at the actual rules, we can see that unarmed strikes are defined as:
Instead of using a weapon to make a melee weapon
attack, you can use an unarmed strike: a punch, kick,
head-butt, or similar forceful blow (none of which
count as weapons).
And just for completeness, here's what Dueling says:
When you are wielding a melee weapon in one hand and
no other weapons, you gain a +2 bonus to damage rolls
with that weapon.
Nothing in there about melee weapon attacks, it's all about actual weapons. Unarmed strikes aren't weapons, so they won't work with Dueling.