You mustn't have watched or read much espionage works, do you?
Hawkeye disobeyed a direct order to kill her when he brought her into SHIELD's protection. The only reason he's still active is that Natasha has proved time and again that he was right in his judgement, but we also saw in The Avengers that Natasha is extremely skilled in playing people, so the risk he took was HUGE, it could've gone really bad for both him and SHIELD. You can bet Fury didn't take her in all that happily initially.
I have. I'm a big fan of the Bourne books. I recommend checking them out. Different than the films but still great.
And while I agree with all you said, it doesn't address anything I pointed out in my post. Yes, taking Natasha in was dangerous, and yes, it was a risk. However, Natasha was not a member of an organization dedicated to destroying everything that Shield was. Natasha was not undercover for years inside Shield, learning Shield intel, and working to undermine the organization. Natasha was a skilled spy/assassin working for another government agency, and she was captured, and then offered a choice. Yes, she was an enemy, but one that was more "out in the open" so to speak.
Think of it this way. It's the difference between these two types of situations:
You're a business owner. You routinely interact with an employee of another company who is in direct competition with your own. Despite this other employee being the direct result of their company beating yours out in multiple sales areas, you can respect their skill, and despite the fact that the other employee is dedicated to their company, you eventually campaign to hire them for yourself.
As opposed to having an employee working for you for years, who is obviously skilled and very talented, only to learn that they have been selling information to your opponent company the entire time they've been hired.
Both situations are similar: You have a person who is hurting your company. However, the person in the first situation is generally looked upon as being more "honorable."
And that's essentially the difference between the two scenarios. Ward was the "dishonorable" spy, while Natasha wasn't. At least, as far as we know. The details of Natasha's induction to Shield is still murky. In addition to that Ward has shown some mildly delusional and clearly sociopathic views at best. His reasoning for why he did what he did is clearly warped, and he has an unhealthy obsession with a member of his former team.