You cannot draw a dagger as part of the attack
The only things that can be drawn as part of the attack are the ammunition used for weapons (such as crossbows) that have the ammunition property. From PHB (pp. 146-147):
Ammunition. You can use a weapon that has the ammunition property to make a ranged attack only if you have ammunition to fire from the weapon. Each time you attack with the weapon, you expend one piece of ammunition. Drawing the ammunition from a quiver, case, or other container is part of the attack.
Since a dagger does not have the ammunition property, it must be drawn using your free object interaction (PHB, p. 190) or your action (to take the Use an Object action); from PHB, p. 193:
You normally interact with an object while doing something else, such as when you draw a sword as part of an attack. When an object requires your action for its use, you take the Use an Object action. This action is also useful when you want to interact with more than one object on your turn.
As you note, this means you cannot draw and throw all three daggers in the
same turn. At best, if you started your turn with two daggers already drawn, you could then throw both of them (using Two Weapon Fighting), use your one free object interaction to draw another, then throw it (as your second attack as per Extra Attack).
The "Thrown Weapon Fighting" Fighting Style, as presented in the Variant Class Features UA, includes the following:
You can draw a weapon that has the thrown property as part of the attack you make with the weapon.
As you note, it goes out of its way to point out that you can draw the weapon that you wish to throw as part of the attack that you make to throw it. This is because, ordinarily, you are not able to drawn a weapon as part of the attack you make with it unless it is ammunition (which isn't actually a weapon either, it's the ammunition you use with a weapon that you're holding).