I honestly didn't even connect Natasha's revelation about her sterilization to her "I'm a monster" thing, just because it seems so obviously nonsensical. I have no problem believing that Natasha thinks she's broken, but, you know, I think that has more to do with the assassinating and what she perceives as an inability to be an authentic person and so on than whether or not she can have biological children.
OTOH, I have to admit that she did go straight from one topic to the other, so they're clearly linked on some level in her head. And what just sprang to mind was Bucky, and his metal arm, and the various ways that fandom and fic handle Bucky's relationship to the arm. One of those strains of thought has Bucky feeling twice damaged, by the loss of his original arm and by having a new one forced on him. And I know we don't have his backstory and Natasha's linked, but they are in comics canon, so...
Natasha doesn't think that an inability to have biological children makes someone monstrous. Natasha feels damaged because she was physically violated and - I say this as one of a number of ways that a person who cannot have biological children might feel, not as an objective fact - broken. It's symbolic for her. I know that in canon Natasha is physically enhanced; we don't have any particular reason to think that's true in the MCU other than her general ridiculousness, which Clint shares so until we're told otherwise I'm assuming she's just awesome rather than juiced up with something. So the (let us assume irreversible) physical representation of what the Red Room did to her, in the sense of her bodily integrity, is her sterilization.
I get the general tiredness with reproductive horror; it usually isn't one of my buttons, but it does get a little old. But I don't find it implausible that Natasha might combine her feelings about her forced sterilization with her feelings about what the Red Room made her into and come out with some unhealthy ideas about what it means about her that she's been sterilized, and so I'm going to go ahead and read the scene that way. Because, as I said at the beginning of this post, the idea that Natasha would think that being infertile makes a person monstrous is nonsensical and terrible and I'm going to need another interpretation.
Also, Natasha Romanoff is a Big Damn Hero. She has such an astonishing heart, to have gone through the things she's gone through and been who she's been and managed to become the woman she is now. That's what I love about her.