I know, finally! But yesterday afternoon was surprisingly busy at work.
I think the problem is that Mai/Zuko doesn't exactly happen "offstage" and that's why the parallel doesn't work. I suppose if you posit that they were not dating pre-banishment but were only sort of interested in each other, so they don't really "get together" until after that, you can say it, but in a way that doesn't really work--how can Mai work up that much love for Zuko when she's dated him for at most a few months after he's been gone for over a year before that?
Really, the problem with M/Z is that Mai as a character and Mai/Zuko as an established relationship is introduced so late in the show. And the "this is why I love her" stuff from Zuko to Mai doesn't really become explicit until Boiling Rock. (Her to him is clearer in The Beach, but it's played rather subtly while many other loud things are going on.) So by the time you get the explanation, it's later than one had hoped for the viewers to really be invested in this relationship. It's why I sometimes say that those early M/Z shippers got lucky, because they liked Mai, and while you can see that narratively there isn't much other reason for her to be there, we all know that most TV show romances don't hang together that well.
As for H/G, if we had seen even one of those "several sunlit days" in HBP the whole thing would have been more convincing. Also, and this may be very pumpkin pie-ish of me, but I really do feel like any girl that was made for Harry wouldn't have let him pull that silly breaking up with her stunt at the end of HBP. The larger problem with H/G, it seems to me, is that JKR sets up a universe in which girls do things, and then gives Harry a girl who is a cliched narrative prize who doesn't do much of anything at all. It's very Amy/Laurie in a lot of ways--a boy (Harry/Laurie) might be close friends with a girl that is clever and capable (Hermione/Jo), but he wants a traditional girl (Ginny/Amy) when it's time to take a wife. It's no shock that a crazy large percentage of H/H shippers were disappointed Jo/Laurie shippers back in the day--or Anne/Gilbert lovers.
Mai/Zuko is very different, and really so is Aang/Katara and Sokka/Suki. All three of those boys like that their girls can kick a lot of ass, and both Sokka and Zuko say so specifically; Sokka and Katara's father openly admires Suki's skills and praises his daughter, so this is a whole "girls kicking ass is awesome and also hot" universe.
But setting that aside, I think it's much tougher to put forth the arbitrary sex part in a teen romance than in a more adult one, not because oh, think of the children, but because adults can be more knowing about sex--instead of being all, "omg, what is this?" they can be "I am clearly hot for you." And in a sense, M/Z is more of an adult relationship, more openly "I am hot for you." And Aang's feelings about Katara are clearly sexual, as Aang keeps staring at the super hot Katara. (One other problem is that in having Aang as the youngish star of a Nick show is that Katara couldn't really do the same, and I can't decide if that was the fault of Nick or Mike and Bryan being boys and not getting it.) Harry doesn't really understand his chest monster; Aang really understands his. Aang knows exactly what he wants. But Harry as a character is so much less in touch with his emotions and his instincts, is so much less intellectual and wise than Aang, that he never knows what he's doing, and that also makes the romance really unsatisfying.
So, I would say that these two ships do not happen offstage in the same ways, and that's your parallel problem right there.
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I think the problem is that Mai/Zuko doesn't exactly happen "offstage" and that's why the parallel doesn't work. I suppose if you posit that they were not dating pre-banishment but were only sort of interested in each other, so they don't really "get together" until after that, you can say it, but in a way that doesn't really work--how can Mai work up that much love for Zuko when she's dated him for at most a few months after he's been gone for over a year before that?
Really, the problem with M/Z is that Mai as a character and Mai/Zuko as an established relationship is introduced so late in the show. And the "this is why I love her" stuff from Zuko to Mai doesn't really become explicit until Boiling Rock. (Her to him is clearer in The Beach, but it's played rather subtly while many other loud things are going on.) So by the time you get the explanation, it's later than one had hoped for the viewers to really be invested in this relationship. It's why I sometimes say that those early M/Z shippers got lucky, because they liked Mai, and while you can see that narratively there isn't much other reason for her to be there, we all know that most TV show romances don't hang together that well.
As for H/G, if we had seen even one of those "several sunlit days" in HBP the whole thing would have been more convincing. Also, and this may be very pumpkin pie-ish of me, but I really do feel like any girl that was made for Harry wouldn't have let him pull that silly breaking up with her stunt at the end of HBP. The larger problem with H/G, it seems to me, is that JKR sets up a universe in which girls do things, and then gives Harry a girl who is a cliched narrative prize who doesn't do much of anything at all. It's very Amy/Laurie in a lot of ways--a boy (Harry/Laurie) might be close friends with a girl that is clever and capable (Hermione/Jo), but he wants a traditional girl (Ginny/Amy) when it's time to take a wife. It's no shock that a crazy large percentage of H/H shippers were disappointed Jo/Laurie shippers back in the day--or Anne/Gilbert lovers.
Mai/Zuko is very different, and really so is Aang/Katara and Sokka/Suki. All three of those boys like that their girls can kick a lot of ass, and both Sokka and Zuko say so specifically; Sokka and Katara's father openly admires Suki's skills and praises his daughter, so this is a whole "girls kicking ass is awesome and also hot" universe.
But setting that aside, I think it's much tougher to put forth the arbitrary sex part in a teen romance than in a more adult one, not because oh, think of the children, but because adults can be more knowing about sex--instead of being all, "omg, what is this?" they can be "I am clearly hot for you." And in a sense, M/Z is more of an adult relationship, more openly "I am hot for you." And Aang's feelings about Katara are clearly sexual, as Aang keeps staring at the super hot Katara. (One other problem is that in having Aang as the youngish star of a Nick show is that Katara couldn't really do the same, and I can't decide if that was the fault of Nick or Mike and Bryan being boys and not getting it.) Harry doesn't really understand his chest monster; Aang really understands his. Aang knows exactly what he wants. But Harry as a character is so much less in touch with his emotions and his instincts, is so much less intellectual and wise than Aang, that he never knows what he's doing, and that also makes the romance really unsatisfying.
So, I would say that these two ships do not happen offstage in the same ways, and that's your parallel problem right there.