Not necessarily. The reason being that "balance" can mean so many things.
Certainly, if you make a claim about balance, you should settle on a definition, first.
For instance, I use the definition:
A quality games possess to varying degrees, marked by a sufficiently or even surfeit of choices available to the player that are both viable and meaningful.
I'd think a game meeting that definition of balance would probably do a fair job of delivering 'genuine choice.'
For example, let's take weapons. Let's balance it so that all weapons do the same damage. Every weapon now
does 1d8 damage. This is balanced. It also renders any choice meaningless.
Which isn't balanced.
For a functionally identical case of imbalance, consider a game that presents twice as many weapons as that first hypothetical, a dozen of them do 1d4 damage, one does 1d8, and the rest do 1d3. Weapons have no other game-mechanical qualities in this system, just the damage die.
In both games, everyone using a weapon runs around doing 1d8 damage. In the first game, everyone's using the weapon their character concepts called for (assuming it's on that shorter list), in the second, everyone's using the exact same weapon.
Balance does not automatically mean more meaningful choices
You can define it as the ability to place the game book in an upright position without it falling over, if you like, it just wouldn't be a useful quality on which to judge a system.
I think [MENTION=57494]Xeviat[/MENTION]'s description is a good goal in general. Choices should matter, and the rules should strive to make them matter.
It's a very similar definition to my own, because we're both looking at the same quality, if in slightly different ways. When some choices are too bad, they stop mattering, and when one choice is too good, all the alternatives stop mattering. Strict superiority is the clearest case, if one choice does everything another does, does all those things better, and does more beside, and has the same or fewer limitations, and the same or more advantages, the other choice might as well not exist, it's that "illusion of choice," and not even a very good one. Strict superiority isn't necessary to make a choice meaningless or non-viable, either, just clear and broad enough superiority (that's where subjectivity comes in, not in whether there is imbalance, just in the individual's tolerance of imbalance).