It was not the case. She was translating someone else’s article, and it does not seem she did it under direction or supervision.
> so many people assume it was her work and not Babbage's.
What she did was quite common. She had ideas about the thing she was translating and thus added them as notes. All fairly straightforward.
> the obsession with attributing the first ever computer program to her seems entirely ideologically motivated.
To me the obsession that some people (not you, but some definitely do and use the same arguments) have with bringing her down is entirely ideologically motivated. She was recognised for a long time, and while there are discussions about exactly who was first and such (as there always are when discussing History), her role was mostly uncontroversial. Also bear in mind that calculator and then programmer were women’s jobs until some point in the 2nd half of the 20th century. Having a woman write code was not controversial before the establishment of the bro culture.