It's a shame the author falls a little foul of the very thing he criticizes - out of a wish, perhaps, to needle individualist liberals with UnHerd-style communitarian-conservative talking points.
But let's ask some more questions.
The positive correlation between income and number of children, within education bands, is intriguing - and yet doesn't explain why policies intended to be pro-birth - generous financial support for young families, for example, seem to be ineffective. Even accepting that education does suppress the birth rate relative to income, increasing income ought to fix that.. but doesn't.
Perhaps there's a different story here, that income and number of children within a given education bracket isn't causal - that a third factor is behind both - or even that the causal arrow points the other way: if you've got lots of kids, you _have_ to take that higher-paying job, because you need to keep them fed and housed.
And there's an unspoken assumption that education = individualism, and that these are antonyms to communitarian-conservatism. Which isn't exactly true, and we can look at societies which sustain relatively high levels of both.
Denmark being a case in point. Its international reputation may be free-living liberal-Scandi, but it's communitarian, place-based and in many ways conservative. Not in the obvious way along Islamic or shouty Evangelical lines - morals around sex and alcohol are relatively relaxed - but in most other respects it's a distinct, cohesive and traditionalist culture, one which places more value on faith and nation than many.
They've high income, strong child support policy, and their birth rate is.. not much better than avowedly secular-liberal countries nearby. And a disproportionate number of kids are born to immigrants, which of course upsets the traditionalists.
But again, look for third factors. Do immigrants have more children because they're from "that kind of culture", or does the sort of innate drive which motivates somebody to pack their bags for a new life in a new country (by no means the easy path) also motivate them to have a family?
Or, flip that around, is there a "drag" factor affecting people who don't emigrate/immigrate, and that's suppressing "native" births? I certainly know more than a few people for whom that appears to be a thing, they're just kind of lightly anesthetized to life.
But I've also heard it said that some folks have always been that way, and that what's driving lower birth rates is more that on one hand fewer unplanned kids are being born (because contraception, and vastly lower teen pregnancy rates), and on the other, the women who do have kids (which has never been anywhere near "all") are often having one or two less than they'd ideally like for economic reasons.