Let's have a look at the fertility rates of Poland, Hungary and Greece, all have a subsidy on birth/children. According to your logic, subsidy = more children. Did we see an increase?
Poland: no
Hungary: yes
Greece: no

Is it therefore that simple? No. Declining fertility rates has many reasons. Affordability and cost of living is a major factor and we'll agree on the reasons why costs of living has risen. Back in the day kids were an asset, they could work on the fields. Now having more kids than necessary becomes a financial burden. Advanced medical abilities creates an option to have less kids because most of them become adults. Why would you want to have more than you need/want? There is simply no need.

Also, you don't need a family to have a kid, it's not complicated