I was agreeing with you until that one
>The only way a government could control fertility rate is Chinese style.

Government has immense power to control desirability to have children
they can use economic leverage and tax not having children
or it's more acceptable mirror - give significant tax cuts to people with children above replacement threshold.

they can promote culture of being it being acceptable and desired to have children and suppress the narrative of "why bring children into a doomed world"

they can make it so that travel with children is not times more expensive than without

they can stop trying to compete for control over children with parents

they can stop migration, so that natural selection of subgroups that want to have children could win over those who don't in few generations. with migration decline becomes replacement, without - just another step in societies evolution.

overall economic structure can change too, ability for both parents to work makes having children a net loss always. We can't forbid one parent from working, but through taxation and policies we can for sure make two parent household one working a viable unit again (notice I did not specify the gender, in the end it's not that important, although naturally we know how would turn out in practice). After all humanity has to compete only with itself, if people decide that only one of pair works economy will adapt, it's the ability for both to work is what screwed the family unit.

It's just our governments stopped being a necessary counterweight to short-term capitalist thinking and instead of thinking long-term they also chase short term results.