I'm curious which classes you think could benefit from this. I think it's ok at level one, it seems subpar after level 5 when most people get multiattack and a feat. I actually think that's a good place for racial abilities though; they should be flavorful for your first few levels, but to underwhelming compared to the items and equipment you later discover.
And to clarify, since you say "if", it's always an unarmed strike, 1/short rest it can be used as a bonus action.
Barbarian, Fighter, Paladin, Monk, Rogue, Bard... basically anything that is going to be in melee range and is likely to lose hit points.
I mean, I figure the closest thing we get to this is races that grant one a cantrip or 1st level spell.
Is dealing 1d6+Str damage on a melee attack in addition to everything an otherwise identical member of the class would normally get to do during that round at least as good as a cantrip or first level spell? It seems to me it most certainly is! Now... is it as good as the Dragonborn breath ability which is another comparable racial ability? I don't think so, BUT... I understand that Lizardfolk might have a bit more going on while that singular ability more or less defines Dragonborn completely.
I mean, if I were designing it then I think I would have been very hesitant to put the ability on there and I would have put some thought into whether I can have the damage scale with level... maybe start off at 1d4 damage at 1st level and get better at roughly the same levels as cantrips increase in power.
But I have already gone quite a bit on record on saying that I feel a lot of the races in this book really aren't as finely tuned as they should be-- like, they are close, but enough under par that they still fail at being ideally balanced and a lot of the abilities have seeds of good ideas in them but needed to be reworked just a little more to get them to where they needed to be.
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You don't need a free hand for an unarmed strike attack. Kicks and headbutts work fine.
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[FONT=Verdana, Arial, Tahoma, Calibri, Geneva, sans-serif]You are completely incorrect. You have demonstrated a common pitfall in understanding of how D&D combat works.[/FONT]
[FONT=Verdana, Arial, Tahoma, Calibri, Geneva, sans-serif]The damage from unarmed attack is not the damage you do by dealing a single strike with any random part of your body with no other consideration. You cannot ball yourself up and roll and say "I am hitting with my head and two arms and two legs, so I deal 5 damage!" that's not how it works.[/FONT]
[FONT=Verdana, Arial, Tahoma, Calibri, Geneva, sans-serif]Unarmed attack is the damage you can do within a 6 second span of time without having any weapons in your hand. Yes, ultimately this means that it COULD be a head-butt or a kick or something. But you know.. there is a reason why you don't see MMA fighters walking into the ring with their hands full even if they intend to use primarily kicks as their offense. You do need to have your hands free and unoccupied in order to deal unarmed damage to the best of your ability.....[/FONT]
[FONT=Verdana, Arial, Tahoma, Calibri, Geneva, sans-serif]I believe you can't even be holding a shield at the same time. (Though, if you have a shield, you should be bashing them in the face with that rather than any body part)[/FONT]
[FONT=Verdana, Arial, Tahoma, Calibri, Geneva, sans-serif]Though, I guess if one is dealing only 1 damage, it really hardly matters.[/FONT]
[FONT=Verdana, Arial, Tahoma, Calibri, Geneva, sans-serif]And this whole concept has always been screwed up by Monster Manual entries that list each claw and the bite as separate attacks. It is perfectly understandable how one winds up with the misinterpretation that unarmed strike is the damage one does with a single punch.[/FONT]