Gender, race, sex preferences, or political beliefs are very different from being anti-vax.
Being anti-vax is evidence of low intelligence. So no, I would not want someone with low intelligence to handle or assist my baby's delivery.
But the most obvious reason is that they directly endanger mothers and babies in the delivery ward because unvaccinated people are more contagious and carry higher viral loads.
Anyway, for a pregnant women of 30 years old with no comorbidity, the risk is very low. A mother giving birth is much more at risk of medical malpractice or general risk due to the various treatments, such as epidurals, ocytocin, episiotomy and so on, that are often not really useful but applied anyway.
Unless they've already had covid of course. In which case the situation is flipped on its head. How many nurses do you think may have already encountered covid prior to vaccines even becoming available?
They are not making that argument. They are simply anti-vax.