could this work?

No. All weapons with the "heavy" property have the two-handed property as well. Here's exactly what each property does (PHB 147):

Heavy. Small creatures have disadvantage on attack rolls with heavy weapons. A heavy weapon's size and bulk make it too large for a Small creature to use effectively.

Two-Handed. This weapon requires two hands when you attack with it.

The heavy property itself isn't a problem here, but the two-handed is. Since all heavy weapons are also two-handed, you must use two hands to wield them. The enlarge spell doesn't give you the ability to wield a two-handed weapon in one hand, even if that weapon wasn't enlarged with you. Your DM might allow it, but it's certainly not part of the Rules As Written.

So what can you dual wield, then? Here's the rules on two-weapon fighting (PHB 195):

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 one hand. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.

Based on this, you can only dual wield light melee weapons. You don't even need the two-weapon fighting style, since all the fighting style does is allow you to add your ability modifier to the bonus action attack. There is a way to get around the "light" limitation, namely the Dual Wielder feat (PHB 165, emphasis mine):

You master fighting with two weapons, gaining the following benefits:

• You gain a +1 bonus to AC while you are wielding a separate melee weapon in each hand
• You can use two-weapon fighting even when the one-handed melee weapons aren't light
• You can draw or stow two one-handed weapons when you would normally only be able to draw or stow only one.

This feat allows you (among other things) to use any one-handed melee weapons for two-weapon fighting.

In conclusion, you're not going to be two-weapon fighting using greatswords, even if you're enlarged. The closest you're going to get would be two longswords with the Dual Wielder feat

Side note: If you want to do two-weapon fighting purely from an optimisation perspective, you may want to reconsider. It is generally accepted as one of the weaker playstyles, as it uses your bonus action to be effective (needed to cast smite spells on a paladin, for example), and still can't quite compete with great weapon fighting, especially if you factor in the Great Weapon Master feat. It doesn't fall completely behind other playstyles, so if you want to do it because of a character concept, go ahead. But if it's exclusively for optimisation, it might be better to just use a single greatsword.

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.