Max Seidman suggests that game designers should think about balancing a game in terms of ensuring meaningful choices rather than ensuring fairness. As he defines it, "Balancing is the act of tweaking numbers (resources, probabilities, options, etc.) to ensure that players always have at least two interesting and equally appealing choices during their turns."
As I see it, there are two ways of making a choice not meaningful. One is to have a clearly dominant option. Seidman spends a lot of time on this aspect in his post, discussing the ways that designers can prune away "false choices" where the correct strategy is obvious, and ensure that there are 2-3 equally appealing options. The other way to make a choice not meaningful is to make the options not matter. If two options are equally appealing, but things will probably turn out the same regardless of which one you pick, then the choice isn't that meaningful either. It has to be clear to the player that the choice will affect the outcome of the game, and the connection between the choice and the outcome needs to be clear. The test of choice meaningfulness, I think, is that it's possible for a player to look back at their choice and say "dammit, I should have done X instead of Y!" If the choice is meaningless in the first sense, nobody will have done Y in the first place. And if it's meaningless in the second sense, players won't think that doing X would have changed anything (because they were destined to fail/win anyway, or because it's unclear whether the X-vs-Y choice made any difference).
This second aspect of choice meaningfulness brings the issue of fairness back in. Unfairness tends to make choices lose meaning in the second way described above. Imagine a game where the first player has a substantial advantage simply by virtue of going first. If the other players recognize this, then their choices will become less meaningful. They can start to feel like it doesn't matter what they choose, as they'll probably lose anyway. Similarly, the player with the unfair advantage will stop worrying about their choices, because they know they can easily coast to victory.
This is why fairness can be so important in a purely strategy-based game -- that is, a game where the only goal is to achieve a concrete victory condition. So we see excruciating efforts at fairness in classic abstract games (e.g. Go, Scrabble) and heavy Euro games (e.g. Agricola, Village). When a game is "unfair," there is usually some other non-strategic goal that makes choices still meaningful. This might be the case in a very atmospheric, theme-heavy game. For example, a horror themed game might make certain players probably doomed, but their choices are still meaningful because they help define the nature of the horror. I might be dead either way, but it matters to the feel of the game how I choose to die. The extreme case of this is a heavily narrative-based roleplaying game like Fiasco. In Fiasco there's no effort at all to "balance" character abilities in terms of making those characters able to "win." Choices are meaningful because of how they influence character development and how they produce situations that the other characters will have to react to. A good Fiasco setup is balanced in the sense of giving characters multiple appealing paths to interestingly different story endings.