dnd 5e – Should unarmed melee weapons like fists and claws have the *finesse* and *light* traits?

In DnD 5e you find under Melee Attacks:

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). On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes.

Also Martial Arts for monks, and Claws for tabaxis and tortles deal with unarmed strikes.

All use Strength for the attack and damage roll, with the exception of the Martial Arts for monks, which accepts Dexterity too.

Should “unarmed strikes with these natural weapons” have the finesse and light traits, because wrestling and boxing are not only based on strength but also dexterity like wushu or bujutsu, and there is no reason to use two daggers but can’t do so with claws or fists, feets etc.? The monk still has more damage and would gain two-weapon fighting with its body. So it would not unbalance in game, but give the tabaxi and tortle non-throwable natural daggers.