Is your idea instead that after she was brainwashed by the Silence in her earliest years at Greystark Hall and then at Jefferson Adams Hamilton in Florida, she escaped from the Silence, got to New York six months later where she regenerated, and then somehow time-jumped from 1969 to 1990 or so to grow up with Amy and Rory?
Yes. I can't see Kovarian letting a ridiculously complicated and important Time Headed Human Weapon run around in New York and get so ill that she actually regenerated. Even the Doctor couldn't find her, and I'm sure he tried. How she got to Leadworth is never explained, but it's possible that River (post LKH) told him where she'd been when she was tiny and he somehow picked her up & set her up with a new identity near her parents. Or she could just have lived her way to them, deliberately staying young. Oh! And there is a hint in an interview with Amy (extra in Summer Falls) that she and Rory tried to find her after they got stuck in New York. So it's possible they found her, and looked after her, until she could go to Leadworth. In either case, she was removed from the Silence.

When she popped back up as River Song at Luna University, it was easy for the Silence to go pick her back up. But as Mels Zucker? No, I don't think Kovarian ever knew where she was.

I like the idea of her only being tortured for her first two or three years much much better than her being tortured for her first twenty-five, obviously.
I never knew people thought the latter! How awful.

And up until Mels starts trying to kill the Doctor in Berlin in 1939 there might have been time for Eleven to find River and change the story, but once Mels kills him the story and therefore the parts of it up until then, including the young-childhood brainwashing, are fixed?
Pretty much, yes. Before Mels crashes into the story, he can save her. But once he knows, it's fixed.