I think for it to be in the job advert, as the title of the role, shows that either the organisation itself, or the hiring manager, has no respect for women and is not considered truly equal. You would always be starting from a place of being judged as lesser. Very telling, when you need to make judgements about whether you want to apply for a job or work for a company. All you can go off is the job advert and their website. Big red flag for me.
Separate from the issue of whether or not the feminine terms should be maintained or absorbed into the existing male/neutral ones, I think that's a massive stretch.
The very fact that they have advertised in a way that (they obviously believe) covers both sexes, surely they have done the exact opposite. It makes as much sense as saying 'he/she' when the sex of the successful candidate cannot yet be known is judging women as less equal and disrespecting them; or, as is sometimes seen, 'she/he' is saying the same about males.
It may well be that the person who wrote the advert is older/old-school/old-fashioned and, to them, manager suggests a male, whereas a female in the same role would be a manageress. Yes, there are logical reasons for possibly bringing the matter up with them and challenging their perspectives and use of language, but this might be their way, from their own upbringing and 'their own time', of deliberately stressing that they aren't only looking for or prioritising a male appointee to the job. They might have gone out of their way to make it crystal clear that they're looking for the best person for the job, regardless of whether they are male or female, and then people have seized on this and used their attempt at inclusivity and non-male-norm assumptions to accuse them of the very thing they were at pains to avoid.
Incidentally, I can't ever recall seeing an advert saying 'waiter' wanted (unless it's one of these seedy 'waiter-in-the-buff' type things for tacky hen nights) - only ever waiter/waitress, waitress/waiter or waiting staff. I agree with the PP that it's important not to inadvertently limit women's choices and freedom of expression in life, by eliminating all specifically feminine references and defaulting to male terms for all. Caroline Criado Perez' excellent book was a timely klaxon call to remember that 'male' does not equal 'person', but then, linguistically, I think there's almost a danger of undermining that message when we demand that anything specifically referring to women be banned/frowned on/deemed belittling, even when there's no suggestion of that intent by the user at all.