This is just not accurate. There are 4 finesse melee weapons and all of them except Scimitar are outclassed by a non-finesse weapons.
Short Sword: 1d6, light, Vex, Slashing. Hand axe is also light, Vex, Slashing but it only requires simple weapon proficiency, can be thrown and is less expensive.
Dagger: 1d4, light, Nick, thrown. The Light Hammer has the same statistics and mastery but with a better damage type
Whip: 1d4, Slow, Reach. The main thing this has going for it is reach, but the 1d4 damage is dwarfed by other reach weapons that are all 1d10.
Rapier: 1d8, Vex: Rapier is outclassed by a host of melee weapons. If you are using a Shield War Pick, Longsword, Morning Star and Flail all do the same damage and have a better mastery for use with a shield and Flail and Morning Star also do bludgeoning damage which is a better damage type to boot. Add Trident which has does identical damage with a Thrown capability. If you are not using a shield then War Pick, Longsword, Warhammer, Battleaxe, Trident, War Pick, Glaive, Halberd, Greatsword, Greataxe, Pike, Javelin and Maul all do better damage than a Rapier.
Scimitar is the only finesse melee weapon that is not outclassed by one or more non-finesse counterparts.