I believe you're suggesting (correctly) that a prediction algorithm trained on a data set where women outperform men with equal resumes would have a bias that would at least be valid when applied to its training data, and possibly (if it's representative data) for other data sets. That's correct for inference models, but not LLMs.
An LLM is a "choose the next word" algorithm trained on (basically) the sum of everything humans have written (including Q&A text), with weights chosen to make it sound credible and personable to some group of decision makers. It's not trained to predict anything except the next word.
Here's (I think) a more reasonable version of your hypothesis for how this bias could have come to be:
If the weight-adjusted training data tended to mention male-coded names fewer times than female-coded names, that could cause the model to bring up the female-coded names in its responses more often.