I think it might be helpful to consider where the trope is being viewed from. By which I mean, it's creation is from a Western standpoint and therefore seeing an Asian woman in film during this time meant you would have the three choices of subservient caretaker, being overly sexualised, or a demonized woman. In the case of the demonized Dragon Lady, the problem lies in how she's constructed. She's shown to have a type of power, but the only way she can possibly keep it, gain it, or use it, is against others in a harmful way. She's one dimensional as a villain and for the viewer it relates how an Asian, in this case Chinese woman, in these old film, was strictly a monster at her core.
It's sort of like the demonization of Japanese men during the world war - the "yellow scourge," whose only plan was to murder and rape any enemy solider's family, wife, etc., unless stopped first. So when you come at the trope of Dragon Ladies, it's through that same point of view that's meant to show its audience how the Asian women as a whole couldn't possibly have a personality outside of that demonized role unless she was a subservient caretaker who stayed silent, or was "useful" to whomever as a sexual object. The only other option is a sometimes sexless monster, and the creation of this is what falls into a racialized problem.
As a side note, you can also look at the character of Charlie Chan. He's goofy, bumbling, has a terrible accent, and spouts his "fortune cookie wisdom," but the important thing during the era of his creation was that he wasn't threatening to the audience. He was a dumbed down detective and therefore not a threat as an Chinese man. So the Dragon Lady is like the flip side to this. She's meant to show how the Chinese or placeholder Asian woman is a threat unless she's stripped of her sex or agency to be, like I said, subservient or overlly sexualised for the original audience, a Western one. In the case, the supposed Asian woman in these types of films are a pure, Western creation and trope placed onto an entire group of diverse peoples.