This is not correct. There are lots of breakdowns as to why, but the clearest and most obvious is to look at productivity.
We are more productive than we were previously. If we compare how much GDP is produced by an hour of labour over time we see that we are getting better and better at making GDP with each hour of labour. We tend to increase at about 1-2% per year, so if one year you made $100000 of value the next year you would make $101000-102000.
This seems odd as surely this should mean people need to work less or have less trouble affording to live? Well, that is where capitalism comes in. Someone is finding life easier, but it isn't you. It is someone who is rich, someone who owns the means of production, a capitalist. By virtue of owning the company they can take your productivity, the total amount of GDP you produce, and then take a slice of that and pay you.
For the capitalist any productivity gains are just pure profit for them, so they love increases of productivity. They also love reductions in labour cost as that is often the largest cost they can potentially get rid of. From their perspective if they could pay you nothing they would. A good example of this is offshoring. Companied do this all the time to send the labour to somewhere that pays the worker less allowing the capitalist to extract even more value.
So how does this relate to the birth rate? Increasing the population increases the number of people you can extract value from. If you are a capitalist this is good because it also increases competition for the jobs you offer, lowering the labour price further.
The simple fact is we could afford to have way less population growth, in fact a bit of a contraction, without any ill effects if we wanted to. We would just need to give the capitalists less of the productivity per hour of labour. They would need to make less profit. They don't want that so they have made a big effort to make people think it would be bad to have low population growth.
Also, parents should get education before having kids. There should be courses you can do to learn what you need to be a parent, we agree there. It shouldn't be women alone getting that. Men who want to be parents should be able to learn the skills needed to be good parents. Putting all that on the mother is sexist and also just silly. Men would be better served by gaining skills to manage parenting than by outsourcing that duty to women.