I know you said you weren't too concerned about balance, but I'm curious (and others might be), so here's a back-of-the-envelope balance analysis of this. I'm assuming no feats, since heavy weapons are already favored for melee damage when feats are in play (as others have said).
As a baseline, take a longsword, no shield, no fighting style, and consider our build options.
For those with access to a fighting style, we currently have the following options, ordered from most offense-oriented to most defense-oriented (I'm ignoring TWF and Protection styles):
- Switching to a greatsword and taking GWF style gives +3.83 to damage per hit and +0 AC
- Switching to a greatsword and taking Defense style gives +2.5 damage per hit and +1 AC
- Dueling gives +2 damage per hit and the ability to wear a shield for +2 AC
- Sword and board with defense style gives +0 damage per hit and +3 AC
Greatsword with defense seems very nearly dominated by sword and board with dueling style: +0.5 damage per hit does not seem worth -1 AC. I tend to think +1.83 damage per hit is not likely worth giving up 2 AC either. There's also the reach dimension: if you forego a shield, then you get the option of trading 1.5 damage per hit for 5' extra reach. Others may disagree, but in my view, without taking feats (or spellcasting object interaction fiddliness, which favors 2H) into account, on a fighter, ranger, or paladin, sword and board is mechanically superior to using a two-handed weapon.
If you change the greatsword to 1d10 but double STR to damage, you're more or less giving characters who use one a flat +2.5 to damage over what they would have gotten under RAW (assuming 18 STR). So now we're talking about a tradeoff between either +3 damage per hit vs +1 AC, or +4.33 damage per hit vs +2 AC.
My gut says that by the time you get extra attack at least, this tips the balance a bit too far in favor of the greatsword, even without feats.
How about for a character that doesn't get a fighting style? Then RAW you're basically choosing between +2.5 damage per hit or +2 AC. That strikes me as a relatively even trade. But now it would become +5 damage per hit vs +2 AC. And if you're a barbarian, this is an even better deal, since with reckless attack you can squeeze more out of that +5, and increasing your AC is worth less when you are giving enemies advantage against you and when you have resistance to their damage.
What if instead of just making this a blanket property of two-handed weapons, you replaced the GWF fighting style with this?