Telemachos was strong enough to draw the bow, though he refrained. I imagine the suitors who tried and failed including one at STR 14, maybe 16. How about Telemachos at 18, and Odysseus at 20?
Anyways, if a guy with STR 10 is punching Target A, and a guy with STR 20 is punching Target B, they should not do damage at the same rate; assuming both targets have the same HP total, the guy with STR 20 should knock down his target down faster. If unarmed strike is a flat 1 regardless of STR, then that doesn't happen. If unarmed strike gets the STR bonus to melee damage, then the guy with STR 20 does damage six times as fast. Oh, and also in that case, a person with STR 8 or less can't do any unarmed HP damage at all, except on a critical.
Anyways, if a guy with STR 10 is punching Target A, and a guy with STR 20 is punching Target B, they should not do damage at the same rate; assuming both targets have the same HP total, the guy with STR 20 should knock down his target down faster. If unarmed strike is a flat 1 regardless of STR, then that doesn't happen. If unarmed strike gets the STR bonus to melee damage, then the guy with STR 20 does damage six times as fast. Oh, and also in that case, a person with STR 8 or less can't do any unarmed HP damage at all, except on a critical.