People, Americans particularly, overestimate the space a human being needs to thrive. I know several families who are healthy and happy in two- and three-bedroom apartments. They pay reasonable rents uptown or a forty minute subway ride out of Manhattan in Jersey or Queens. That’s much better than almost any metropolis, where sprawl makes the density gradient gentler and thus commute times between cost zones longer.
Firstly, to be perfectly frank, you can get 2, 3 years into your first kid before a studio makes a practical difference over a 2-bedroom apartment. For the first chunk, keeping the crib in your bedroom for ease of care is pretty normal, and once the kid is mobile chances are they're going to be sneaking into your bed to sleep at every opportunity anyway.
Secondly, the average length of ownership of a house in the US is like 7.5 years. A lot of people never make it through their 30 year mortgage because they either move for work, their circumstances change and they need more or less house or their income changes and they can/can't afford a nicer, bigger house than they could before.
So living in a studio well into baby #1 before deciding to trade up to a larger place when it's time to look for the "good" schools or baby #2 is on the way is kind of a perfectly normal way to go about things?
Also, one bedrooms are luxurious. Advice for anyone looking for an apartment in Manhattan: don't waste your time on one bedrooms, look for studios. You pay an unnecessary premium for the extra wall, and it ultimately just makes the space less usable. You can find larger, nicer, better located studios for much less than you'd be paying for a one bedroom.