Tony and JARVIS deal with a lot of war vets, and I have to wonder if--on some level--they both clocked Natasha as 'extreme psych damage', not 'dangerous'. Tony wouldn't have brought Natasha into JARVIS' body if he thought she was going to hurt people for fun, and JARVIS probably can read facial expressions well enough to notice that she still had flat effect as Natalie Rushman, just pasting on a smile so no one would notice. Plus, she actively tried to help out Tony multiple times.

And it's worth bearing in mind: WE know that she could've just asked Tony to take the injection, but SHE AND FURY saw him behaving in ways that were outright suicidal. Medical ethics around slow suicide like that are VERY advanced, and spies aren't trained in it--she screwed up, but her screw-up was in the direction of 'save life now so he doesn't die', and that's a pretty reasonable response to 'this guy is gonna kill himself if we don't intervene' if you don't know what you're doing.