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.
Best Answer
Tl;dr: Nope. Not as your primary action, and not as your Two Weapon Fighting attack. Try using haste instead!
There are two places in the rules for you to consult - the rules for Two-Weapon Fighting in the Combat section of the Player's Handbook:
...and the rules for Grappling directly adjacent to them:
There are two reasons that you can't make an "off hand attack" that initiates a grapple. Firstly, the rules for grapple specify that you can only use the Attack action (note the capital) to initiate a grapple, while Two Weapon Fighting allows you to use a bonus action to make an additional attack with a light weapon. The second reason is that Two Weapon Fighting requires you to be holding a light weapon in each hand, while grappling requires a free hand.
However, you may not have to abandon hope of Barbarian attacking+grappling shenanigans with your high Athletics score - provided you can convince a friendly caster near you to provide you with a cast of the haste spell.
Unfortunately, an attempt to grapple is not a "weapon attack", and so you can't use your Haste action to grapple them. However, there's nothing to say you can't use your regular action to grapple them, and them hit them upside the head with your Haste action.