Yep. I would have suggested a +1 to hit, damage or AC, but AC is far too good to be comparable to even to-hit numbers. Increasing the damage to +2 would just conflict with the duellist style as well. Giving a +2 to damage when using the weapon 2-handed might be balanced, but I'm not sure what a fitting bonus for using it 1-handed would be.
You do have the free hand to use a shield, grab, or whatever.
Thing is, "quicker and more agile" doesn't mean "Strength is irrelevant". The entire reason that a versatile weapon is quicker and more agile in two hands is because the additional leverage makes a better strength multiplier.
Sure, but D&D has an all-or-nothing model for now, with STR being the default, and finesse letting you use DEX, instead. In reality, I assume, both strength and agility (among other things) matter. Games have gone there in the past, IIRC, the in RuneQuest, for instance, STR, DEX, & INT all added to your attack % with melee weapons, as did two other stats, Size, and Power (your mystical connectedness to the universe). Most of them only added a little if they were very high, with STR and DEX being the more significant. With bounded accuracy, obviously, that would be madness.

I've toyed with the idea of some feature or weapon quality letting the wielder add both STR & DEX (or two other specific stats), up to a maximum of +5, which would encourage less Single-attribute specialization in classes like the fighter. It would, though, make it easy to get a +4 or +5 to hit out of the standard array at low level, and so decrease the optimization impact of ASIs (and bonus ASIs are one of the fighter's 'things') unless feats are in play, then it'd have the opposite effect, making taking a feat at 4th while starting with a max stat bonus to attacks more doable.
Reasons I would use a versatile weapon one-handed compared to in two hands:
I want to use a shield for better defence - particularly against multiple people or missiles.
I want more reach - if my opponent is using a slightly shorter weapon, or using two-hands, or in a line-fight.
Obviously the shield helps with defense. A reach advantage is both offensive & defensive, isn't it?
Reasons I would use a versatile weapon in two hands compared to one hand:
I need to hit something much harder.
I'm fighting a larger or heavier weapon or stronger opponent, don't have a shield, and so need the extra bracing when parrying.
I need the extra speed, control and force to either get around my opponent's guard, get through her armour, or both.
Again, both offense & defense, though clearly more on offense.
- Most of these are much more situational and based on specifics such as the weapon or armour your opponent has. They don't translate well into D&D combat mechanics.
Of course. D&D is very abstract that way.
Maybe +1 AC when used one-handed, and +1 to hit when used two-handed would cover it? I'd like something more interesting than flat bonuses, myself, but that'd model what you describe in a preponderance of the evidence kind of way, and it's a clear choice, that synergizes with the option of a shield one-handed, and the larger damage die 2-handed. Defense style already gives a +1 AC, so there's a stacking danger, there, and +1 AC is a big deal since AC currently scales only with DEX and gear. It might be desireable to phrase it in a way that doesn't stack with Defense style (then again, the only ones who get two styles that I recall are Champions, and it's not like they're in danger of dominating the game as it is).