So, is it safe to assume that "advanced jobs" is a natural sounding, commonly used phrase?
I wouldn't go that far. I agree with David that "highly technical jobs that need expertise" is the most likely intended meaning of "advance [sic] jobs". The combination of repeated typographical errors on the part of the author and the appearance that Dr Bellock was speaking extemporaneously leave room for doubt both as to what was actually said and what was actually meant. (I believe we can rule out the "lost in translation" factor. Dr Bellock was born in Berkeley, California, and is a native speaker of English.)
I don't find "advanced jobs" to be either natural sounding or commonly used. It sounds like a phrase that came to Dr Bellock on the spot when she was experiencing very similar pressure to the type she was actually researching at the time. I think I would prefer a phrase like "highly skilled positions" or "highly specialized positions" over "advanced jobs".