To give all Sasuke pairings, including SH, Sasuke defense, and to tangentially respond to the above slightly while trying to keep the topic focused , I think Sasuke -does- care about people, in canon specifically Team 7, and Team Taka to an extent, and his later family and it wouldn't be a stretch that it would be the same for Hinata if events set them up that way to be on the same mission and the like. He'd deny it, or be stoic about it, or just overly focused on revenge, but it is there. Or at least, in retrospect, this interpretation has validity.
Put Hinata through simliar circumstances and you have a working theory. The only issue is either approaching it by removing Naruto, and thus really changing Hinata, and really having a darker thing altogether, because Sasuke isn't the best person to seek inspiration from, or a odd last minute fade. I think the former is more effective potential wise.
And therein lies the rub: What is darker for Hinata also is darker for Sakura. One is no less deserving than the other.
By removing Naruto from Sakura's storyline, it really becomes terribly sad and dark. She's no longer a ninja or a medic, she a housewife raising a child alone in a village, with an unrequited love who will never return. She's lost everything there. She's lost her own goals, as well as her goal of bringing Sasuke home. And then, to add insult to injury, her own daughter, despite being raised lovingly by a single mother, judges her with 'you're not my real mom.' So it's implied that no matter what her sacrifices have been, Sakura's also not so good at the only job she's been reduced to.
Compare this to Hinata, who's shown glowingly as a mom looks almost as young as her daughter. Like she's a playful babysitter/older sister rather than a worn out housewife. (Thinking of the two manga images here, of Sakura cleaning and Hinata at the graveside.)
Anyway, only the audience judges Hinata to be the more deserving of the two girls. But they are really the same. Put Hinata into Sakura's situation — living in a loveless home with all goals stripped away — and the harm done to Sakura's character and storyline becomes really clear.
What is dark for Hinata in theory, becomes dark for Sakura in the reality of the ending.
(Honestly, imagine the wellspring of sympathy Hinata would get if Sasuke left her pregnant and alone in the village, living in an old Uchiha compound, raising a child who didn't believe she was her mom, and spending all her days worrying about bills and dusting cobwebs from corners! People would be weeping for her! But they only shrug and say Sakura deserves it, it was what she wanted. 
)
Back to the idea of SH though, it only works — like all things Sasuke-related — if he undergoes a personal evolution of some sort. It's really not the lack of love on the part of Hinata (or Sakura in the manga version). It's that Sasuke is never shown to return love after he leaves the village, except to show mercy or partiality to Karin. SH works if Sasuke becomes more human and shows concern or regard for Hinata. So Sasuke has to change, not Hinata. And this is the same reason that SS doesn't work in the manga ending, even though Sakura supposedly got what she wanted. But Sasuke never evolved to love anyone other than himself. (And he still hasn't.)
