You're exactly right.  If you ask anyone why people aren't having kids, they will say money, because they want the system to give them money. 
I was with you in the first half, but I don't think the "because money" argument is the result of wanting government handouts.
My grandfather was an Italian immigrant, who came to America right before the Great Depression hit. He only got a 9th grade education (some variants of the story say 6th or 8th; suffice it to say he didn't graduate high school), and then he worked for a defense contractor doing machine work. On one income (admittedly 80-hour work weeks were the norm for that income), on a 9th-grade education, he was able to raise three children, have a custom-built house in a then-semi-rural part of the country (that still had public transit to get to and from work), and still save for retirement.
How much money does it take to raise three kids in a custom-built home and save for retirement within two hours of a major city, and how many jobs make that much money reliably, and how many of *those* jobs can be had with a 9th grade education?
Sure, my parents didn't get designer clothes, and only had maybe three family vacations between birth and adulthood, and they didn't eat out frequently....but while there are certainly areas that twentysomethings can be more prudent about, the median family income in 2024 was $83,730, and that usually involves two incomes. 50 years ago, it was $6,600, or $69,150 inflation-adjusted - roughly a 21% increase. Meanwhile, the median home cost $20,264 in 1964, or $212,314 inflation-adjusted, while median house cost in 2024 was $414,361 - a 95% increase.
I haven't even considered the costs for everything else - Even if you want to argue that families usually had only one car instead of two, they cost more than twice as much now. Even if you want to argue that food is different now than it was then, it's also twice as expensive. Even if you want to argue that stay-at-home moms *were* childcare in 1964, the functional need for two working parents means that child care is a requirement that wasn't a need at the time.
So...yeah - I'm not a government-should-pay-for-everything, socialism-but-real-socialism-not-Cuba-or-USSR-socialism leftist, but the math simply makes it completely impossible for a couple in 2025 who isn't in the top 10% of earners to have the life that our grandparents and parents did - even if such a couple is willing to live in a 1600 square-foot home, drive a single car, go on infrequent vacations, live in an average region of the country, and have a daily diet of rice and beans - and if homes were $45K in 1964 against $6,600 median incomes, you can bet that there would be a lot less people in this country, too. ...and yes, I understand that my numbers are USA numbers while TFA is about Finland, but I'm hard pressed to believe that their numbers don't have a simliar trend.