I don't think this is true. Steady employment causes people to get more kids. Just take a look at Korea, its new administrative capital with most public workers has the highest fertility rate.
Isn't South Korea experiencing a massive demographic crisis right now due to a record low fertility rate?
Yes but the places where this is least the case are those with the most stable employment, I believe Sejong, the location with most public workers, has a whopping 1.2-1.3 tfr, 40% higher than the rest of the country and 80% higher than seoul. Obviously you can't make your whole country public workers but it shows that there's some benefit to stability.