[RPG] treat an unarmed strike as a “Light weapon” in terms of two-weapon fighting?

From the rule you quote:

If either weapon has the thrown property...

Which weapon(s)?

a light melee weapon... [and] a different light melee weapon.

So the weapons of "either weapon" mention are the light melee weapons from the previous paragraph.

You can't throw both darts, because they're not valid weapons for two-weapon fighting.

And neither is a melee weapon, so:

You can't make improvised (dagger?) melee attacks with both.

But doesn't using a dart as an improvised dagger make it a melee weapon? "Improvised Weapons" (PHB p.147-148) says "in many cases, an improvised weapon is similar to an actual weapon and can be treated as such." (emphasis mine)

Nope. Later in "Improvised Weapons" we have "if a character uses a ranged weapon to make a melee attack...." (emphasis mine) The authors could have said "ranged weapon as a melee weapon" or something that indicated some change in the weapon's type, but they didn't.

It's still a ranged weapon, per the table, even when used to make a melee attack.

Yup, this works.

He takes the Attack action, and meets all of the prerequisites of TWF. Check.

He throws a dagger, as TWF allows. Check.

Now, as you describe, he's holding a light, one-handed, melee weapon and can attack with it. He can either make a melee attack or throw it, per the last line of TWF.

If he could not attack because he's only holding one weapon, the last line of TWF would be nonsense. The conditions were met when he chose the Attack action, so he can carry out all of its steps.