Critical Hit from Thrown Dagger

I am trying to find out if a dagger can threaten a critical hit on a 19 when thrown as its crit range is 19-20. I am making a Flying Blade Swashbuckler and am interested in throwing daggers as my method of attack. The rulebook says:

Thrown Weapons: The wielder applies his Strength modifier to damage dealt by thrown weapons (except for splash weapons). It is possible to throw a weapon that isn’t designed to be thrown (that is, a melee weapon that doesn’t have a numeric entry in the Range column on Table: Weapons), and a character who does so takes a –4 penalty on the attack roll. Throwing a light or one-handed weapon is a standard action, while throwing a two-handed weapon is a full-round action. Regardless of the type of weapon, such an attack scores a threat only on a natural 20 and deals double damage on a critical hit. Such a weapon has a range increment of 10 feet.

Another member of my party believes it would not due to that second-to-last sentence:

Regardless of the type of weapon, such an attack scores a threat only on a natural 20 and deals double damage on a critical hit.

Similarly, this would also mean a starknife would not receive its ×3 critical modifier if thrown. Please help me understand why this would or would not occur. I can’t find a clear answer for this anywhere.