Hey,
I'm looking to do similar with my wife in September,
so from my understanding you're being advised you can simply come on a tourist visa, overstay, have the baby, and then use the babies birth certificate to apply for PR immediately, and likely citizenship shortly after, and that the overstay will not be an issue?
I'm always paranoid about immigration stuff, so any info you could share - also about life since you've arrived, maybe getting local drivers license or any experiences in settling in the last month would be tremendously helpful.