I'm not sure if I got your question right but

You already apply the ability modifier to the 1d4 attack with Polearm Master.

This is from the sage advice compendium:

Polearm Master Can I add my Strength modifier to the damage of the bonus attack that Polearm Master gives me?
Yep! If you have the feat and use the Attack action to attack with a glaive, halberd, or quarterstaff, you can also strike with the weapon’s opposite end as a bonus action. For that bonus attack, you add your ability modifier to the attack roll, as you do whenever you attack with that weapon, and if you hit, you add the same ability modifier to the damage roll, which is normal for weapon damage rolls (PH, 196). A specific rule, such as the rule for two-weapon fighting (PH, 195), might break the general rule by telling you not to add your ability modifier to the damage. Polearm Master doesn’t do that.

You don't need the Two-Weapon fighting style to do that, it's already incorporated in the feat.

A Free Hand.

In order to Grapple a target, you need a free hand

In order to cast a spell with Somatic and/or Material components, you need a free hand. (Note: War Caster alleviates the Somatic limit.)

If you want to be able to manipulate objects in the environment around you (without having to sheathe or drop a weapon), you need a free hand.

So, in general, two sorts of character benefit from this loadout.

Gishes

^Slang term for a Spellblade, Swordmage, etc. Someone who fights with melee weapons and magic. If both hands are occupied you cannot cast spells with Material Components. If you do not have the War Caster feat, you also cannot cast spells with Somatic components. This severely limits the spells that you can use.

Thus, a character like a Bladesinger or Arcane Trickster will get a lot of mileage out of 1H only.

Grapplers

You can't grapple if you don't have a free hand, but you only need one free hand in order to grapple. So any character built as a grappler will be best served to use this pairing. You have a free hand to grab your opponent, and then a hand with a weapon in it to pummel them with once you've grabbed them.

Anyone who can't use shields or 2-handers, and wants to save their Bonus Action for something else.

Your statement:

By default (ie. no feats), a character can wield a heavy/regular/light weapon in one hand, whilst wielding a light weapon in their offhand.

is incorrect. PHB 195 says

When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand.

Emphasis Mine. By default, dual-wielders must dual-wield light weapons only...both hands. You must have the Dual Wielder feat in order to use non-light weapons in either hand. Thus, a feat-less character with 1H only can use a higher damage main hand weapon than a feat-less character who is dual-wielding. A Rogue (or other character who cannot use Shields or Two-Handers effectively) who wants a higher damage main attack, and to save their Bonus Action for something else would also be well suited to the 1H only set up.

Summary

Ultimately, yes. If you are a pure melee fighter, not using one of your hands to fight is generally a mechanical disadvantage. Most characters whose primary combat method is melee brawling are best suited to make use of both of their hands when fighting. Whether that's 2 weapons, 1H and Shield, or a 2-hander.

But, if you're a gish a grappler, or you want to save that Bonus Action (and can't use two-handers or shields)...1H and free hand is powerful.