Sorry, that was more me trying to wrap my head around the idea of ranged being better. Let's take an example.
Dwarf plate axe fighter against Human leather bow fighter.
I would say that if it were 1 v 1 in an open field, yes the Human would have the advantage as long as the dwarven fighter had no other means to also bring his attack to a ranged attack. Say he had a throwing axe, not only is his high strength modifier going to help him hit, its going to be easier to hit the human because of his lack of armor (tho his dex may help him a few points.) if you stick that human in heavy armor, he loses that dexterity bonus altogether. I admit that he could simply use Thrown weapons as well but as a human he would then be slower than the dwarf unless he had str 15+. Then his thrown ranged weapons do NOT measure to a great axe/battleaxe w.e melee weapon you want most likely.
I am just not sure where ranged = better
This is because you make suboptimal build choices made out of old habit.
First mistake: player characters do not fight player characters, they should be built towards fighting monsters. (A monster with the Dwarf plate axe character's stats would be a immensely dangerous foe to meet; much more capable than most monsters) In your PC vs PC duel you miss out on very important advantages of ranged fighting. Just to mention one - the ability to reliably reach the next foe when your first attack has downed your current one. While an axe fighter will sometimes find he can't reach the next foe with his remaining movement, this happens significantly less often to somebody with 150 ft range. And even a single missed attack in a short fight tilts the balance squarely in the ranged fighter's favor.
That said, okay so it's a one on one duel.
Either assume both fighters have a +5 in their prime stat, so that the studded leather human has a a base AC of 17. Or don't assume a leather archer, instead assume a half-plate human, also with a base AC of 17.
That throwing axes can use Strength for both attack and damage in 5e is a great addition. But ultimately, thrown handaxes has only a 20 ft effective range, which means your Dwarf's range is only 45 feet.
If the dwarf goes for shield plus battleaxe, I counter the shield with the Archery fighting style and a longbow deals the same damage as does the battleaxe.
If the dwarf goes for greataxe, he does retain an advantage in the basic game. The game is better balanced without feats, see. As soon as you give the Dwarf GWM I give the human SS, which is a losing proposition for you (if only because Cleave is worthless here... this being a 1:1 duel

)
In the end you will find that the Axe Dwarf has an edge, but a small one. And now for the million-dollar question: is this edge enough to balance all the benefits of range that isn't visible in this white-room duel...?
(Not to mention what would happen if I removed my silk gloves, and gave the human a hand crossbow, and both SS and CE feats. But that's an obviously broken combo in my view, so I won't)
Best regards,
Zapp
PS. You didn't elaborate on your assumptions, so just so we're on the same line here:
Even so, melee does more damage per hit. As Dex mod does not impact damage on a ranged attack but Strength mod does impact a melee hit.
That is what my Redux accomplishes. It's a change from the core 5e game.
As well as Melee users typically have higher AC and more HP.
Melee typically have one to three points higher AC. But if they use a shield, they lose much of the damage edge available to melee.
And besides, even if you only gain a single round of being able to attack while not attacked back, that easily swings the entire balance in the favor of range.
Your assumption that melee users have more HP than ranged is not true in 5e.