I'm not really qualified to talk about the psychology behind it, but I've observed that people who have 2 or more kids are usually friends with other people who have kids. And it's not as if they became friends when they already had children; it's more like people in their close friend network from their teens and twenties decided to have kids and the others jumped on the bandwagon.
My wife and I are like this. My wife's friend group from high school, who she's still very close with, consists of 5 women. The first one had two kids relatively early, in her late twenties. Then the floodgates opened, and then two more had kids, then we had two kids, and there was one holdout who I never thought would ever have children, since she and her partner lived artist lives in a one bedroom and were a bit more financially precarious than the rest of us. But, sure enough, she ended up having a child in her late thirties and they still make it work in their 1BR.
I'd say I even experienced this as a man. I have two separate friend groups of men that I made at different times in my life, where each member is a friend with somebody else in the group. In the first group, everyone is a father except one person, and in the other, nobody is a father except me and another person. Not gonna lie, belonging to the "father" friend group may have influenced my decision to want to have children somewhat, as I was one of the last of that group to have kids.
I don't know what governments can do except to make having children not only easy through financial supports, but to also zone and legislate an environment that attracts families that may want to have many children. Make restaurants friendly to large families (tables that accommodate more than 4, doors that can fit a stroller, etc.), make hotels kid friendly (rooms that can accommodate more than 4). Some of those can be legislated at the local level, and others you have to wait for the market to intervene, but anything helps.
There's
a town in Japan that did this with a TFR of 2.8 and where almost half of all households have at least 3 children. This is a huge outlier in a country famously known for its low birthrate.
We may end up being a society where there are "breeding islands" like this - communities that are very kid friendly that attract people that want to have many children and prop up the birthrate for the whole country. Whatever works.