The Trait Description for Fast Shot is as follows:
You don't have time to aim for a targeted attack, because you attack faster than normal people. It costs you one less action point for guns and thrown weapons, and then one less with two handed weapons.
There is nothing written in here about a damage bonus, if it was in a changelog at some point in 2238 I wouldn't doubt it, but chances are One Hander's bonus would be in there too.
And about Heavy Handed, it doesn't actually add "Melee Damage", it adds +2 damage on to the end of Melee & Unarmed attacks. It's not "real" melee damage and thus doesn't actually show during character creation when you pick the trait. The description for Heavy Handed however does tell you that your punches are more damaging, but this is a leftover from the Fallout 2 description of the trait.
In Fallout 2 Heavy Handed was +4 melee damage (actual melee damage, not +2 damage on the end) for -30% crit roll, while in 2238 the change for Heavy Handed was made to make it all about the knockdown strength rolls, its real purpose. Bruiser in 2238 had since taken Heavy Handed's place as the melee damage buff (as it doubles base melee damage), as well as +4 STR for -2 AP.
If we're going by trait descriptions at all in this anyway (which we shouldn't, because that's not how you balance games), One Hander has much more right to +5 damage than Fast Shot just by the ambiguous wording of it.
One of your hands is very dominant. You excel with single-handed weapons, but two-handed weapons cause a problem.
It's so insignificant and tiny, a tiny damage buff to people who chose to gimp their builds by being able to only use One Handed weapons properly, I have no real idea why the code was removed instead of just fixing it, when the exact same conditions apply to two other traits.
At the moment the trait is just a bad version of Small Frame that only works with one handed weapons. +20% to hit allowing you to dump 1 STR point below what the one handed weapon you desire requires, since using a weapon when you're 1 STR below its requirement is -20% to hit. This trait is completely useless and nothing at all sets it apart from the vastly superior Small Frame since it doesn't even have a tiny bonus on it like Fast Shot (which is already good).