Here is my non-spoilery preface: the only reason we have to debate Natasha's representation in Ultron so much is that she's the only female character in a lead role. This film gives us some awesome ladies around the periphery, like Laura, Maria Hill, and Dr. Cho. Scarlet Witch/Wanda Maximoff is more fully developed, but Natasha is the only female in the principal Avengers cast. Because of that, she has to represent everything everyone wants from their empowered lady character -- and since different people find different things empowering, that's an impossible burden. An ideal film would contain many different women whose life and relationships are explored, thus representing the multitude of ways women in the real world choose to live their lives.
If I could change anything about how fandom reacts to female characters, I'd want people to stop automatically saying that a woman who wants a relationship is "reduced" to being a love interest. Yes, in some films, being a love interest can be reductive -- but that doesn't change the fact that many strong, independent women choose to be in romantic relationships, and that choice is not inherently disempowering. Pursuing a relationship can be a powerful form of agency, just like any other major life decision a woman makes. With regard to Natasha in Ultron, I would argue that pursuing a romantic relationship with Bruce Banner demonstrates a lot of agency.
First, Natasha's storylines in the two previous films have nothing to do with men (at least, not in a romantic way). In Avengers, she's arguably the hero. She plays the femme fatale damsel in distress only to mindfuck her captors into giving her information. Thereafter, even though she's the victim of some male gaze camera shots, her role is completely non-romantic and non-sexualized. She fucks with Loki's head. She hijacks a Chitauri speeder. She fights her way to Loki's scepter and ultimately closes the portal. Not one dude even flirts with her, and she expresses absolutely no interest in men. In Cap 2, we see a slightly more laidback version of Natasha. She flirts with Steve, but not in a serious way - it seems more like flirting with people is just part of her personality. Without her spy skills, Steve absolutely would've gotten arrested in the mall, and she's the one who reveals all of Hydra's secrets on the internet.
Both of these films prove that Natasha is perfectly fine without a man, which in turn proves that her pursuit of Dr. Banner in Ultron is her personal choice. Even though the pairing is super awkward on screen, her rationale makes sense: she's surrounded by fighters, and Bruce is a scientist. He's different. He's interesting. They're both working toward redemption, and he might actually have a chance at understanding her complicated past. Some people have argued that Nat is captured by Ultron just so she can be rescued by Bruce, but I would argue that's not a strong reading of the text. First, Natasha is captured while executing a dangerous cargo transfer onto an airplane...during which she actually falls out of said airplane. She knows that this is a risky maneuver, so much so that she explicitly instructs Clint to go on without her. In other words, she thinks this mission is so important that she's willing to sacrifice herself to do it. If she hadn't been there to make sure Clint got the cargo, Tony could never have built technology Jesus the Vision thing and defeated Ultron. Then, she facilitates her own rescue by finding a way to send a radio transmission from her cell. When Bruce shows up, she kisses him and throws him into a pit to make him do what she wants. As always, everyone's mileage varies, but claiming that Natasha is "just a love interest" or "just a damsel in distress" is disrespectful of her choices.
The second criticism I've seen is that Bruce doesn't reciprocate Natasha's feelings. I didn't actually read the situation this way -- he does suggest they run off together, after all -- but there is absolutely nothing wrong with pursuing a guy who ultimately turns you down. Going after a guy instead of waiting for him to come after you is the actual definition of agency. It's not like Nat weeps in a corner when it doesn't work out. She takes some time to feel grief, and then she goes right back to work because her status as an accomplished, independent, professional woman hasn't changed.