I think the song is really about George justifying his unconscionable behavior toward Dot, by hiding behind the Noble Beauty and Sacrifice of art. He's telling himself (and us) that it's okay for him to be a jerk to Dot (and to others) because he has something more important to do than worry about people's feelings -- he has art to create, he has to "finish the hat." And in that light, the song takes on a much darker tint, and also a much clearer purpose in the show. Throughout the song, the title phrase is always part of a larger thought, not standing alone by itself, and we can't ignore the context in which that phrase is used. Sondheim is telling us something very specific here about George, not about art. After all, this is Sunday in the Park with George, not Sunday in the Park with Art.
At the beginning of the song, George says:
Let her look for me - good.
George really believes that Dot is the one at fault here, not him. The main idea of this song is set up quite clearly, here at the beginning (as it is with most Sondheim songs). This is not a song about art; it's a song about blame. George goes on to say that no one can possibly understand the reasons for his behavior. Maybe that's why people always think it's his fault, when he knows it's always someone else's fault.
The first time we hear the phrase "Finishing the Hat," it's not the beginning of a thought; it's continuing a thought. George is saying it would be nice if anybody could understand the act of finishing the hat, could understand his compulsion to put his work above all else, "how you have to finish the hat." In other words, it's not his fault he's inattentive, insulting, thoughtless, rude -- it's his art's fault, because that's what forces him to be the way he is. He justifies the fact that he watches the world rather than participating in it, which by implication justifies the fact that he refuses to play by the rules of the real world. After all, he's not a part of that world, so why should he live by its rules? (And yet, we'll see at the end of Act I that he's not a part of the world of his art either; he is outside of it. There is nowhere he belongs.)
George is aware of the real world, but only as "voices that come through the window... until they distance and die." It's interesting that the world of his art is a world of light, but the real world is "the night." George sees the real world, the people in the real world, as inferior (though we must be careful not to assume George's opinions are Sondheim's opinions). So he refuses to interact with the real world in any meaningful way. He thinks he must keep himself at a distance so he can fully observe it. He thinks "It's the only way to see." The only way to create art, he believes, is to remove himself from the real world -- and thereby ignore its rules and conventions.
Late in the song, he says:
When the woman that you wanted goes,
In other words, if she can't handle it, that's just tough. She knows George's rules coming in, and if she can't live by them, that's her problem and not George's. It doesn't even occur to him that he should change his behavior. That's not even an option. He knows that anyone who gets close to him figures out the most basic truth about him: no matter what he's doing, there's always a big part of him that is not there, not in the real world, not in the moment, a part of him who's standing back, watching, not interacting, not caring, just observing. And that's his justification for being the way he is. It's not because he's a jerk; it's because he's a great artist. He really believes that he must submerge his emotions, he must reject polite society, he must ignore the complaints and needs of those who care about him, because if he gives in, if he allows himself to live in the real world instead of in the world of the hat, he will no longer be a great artist. He thinks his mission as an artist gives him universal absolution. But Dot blows a hole in his arrogance in the song "We Do Not Belong Together." She says to him, "You have a mission, a mission to see. Now I have one, too, George." (She's talking about raising a child.) Dot is saying to him that creating art may be important, but other things are important too. (Interestingly, their two missions are what Marie will sing about in Act II -- "Children and Art" -- the only two things worth leaving behind when you die.)
But "Finishing the Hat" is not about the creation of art any more than Fiddler on the Roof is about the Russian Revolution. "Finishing the Hat" merely uses the creatiion of art as an excuse for George's behavior. His argument is an eloquent one, but it's pure bunk. And perhaps he doesn't even really buy it himself. Yes, some great artists were nasty people, and other great artists were friendly, kind, compassionate people. Creating art is not a legitimate excuse for being a heartless, cruel man. George is asking us to feel sorry for him, poor misunderstood, innocent artist that he is, but his argument is not compelling enough. And perhaps that's Sondheim's greatest achievement with this remarkable song -- it is beautiful, even a little seductive, but we don't accept George's excuse. If we did, if George himself accepted it, then there would be nothing for him to learn, no reason for him to grow, and no reason for the story to continue. I think George knows it's bunk.
Could this song just be about the creation of art, and not all this other stuff? Well, Sondheim doesn't just stop the story anywhere else in Sunday for mediations on related topics. Songs like "Finishing the Hat," "Beautiful," and "Lesson #8" let us see George trying to figure things out, trying to learn. To ignore that is to ignore the fundamental action and structure of the show.