Things aren't quite this simple because developed nations don't act as homogeneous wholes. The trends we see on the international level are followed within nations themselves and even down to the state, county, and city level. For instance this [1] is a graph of US income vs fertility. People who are earning substantial amounts of money are, like you said, simply not not reproducing enough to maintain their population. Yet people people who cannot even afford to feed themselves, let alone any children, are reproducing rapidly. The birth rate of those earning less than $10k is nearly 50% higher than those earning more than $200k per year. And the gradient between these two is somewhat disconcertingly smooth.
Religion is another factor that plays into birth rates. Muslims, for instance, have extremely high fertility rates and that stays true even once they migrate to developed nations. For instance the Muslim fertility rate in North America is about 2.7. It's also why the Muslim population is expected to increase 70% by 2060 whereas the number of unaffiliated is expected to increase by 3%, Buddhists decrease by 7%, and so on. [2] Those number increases are not affected by conversion which plays a roughly net zero role in Islam with about 25% of people leaving the religion, and 25% joining it.
So what you're looking at is more of a demographic switch than an economic one. People who are highly educated, secular, and high income will gradually die off, failing to replace themselves. At the same time individuals of lower income and those who place less value on education and more on religion will continue to increase in numbers well beyond replacement. Basically, so long as there is at least one group within a nation that consistently produces in excess of replacement, they will eventually replace any and every group that does not. And, in the longrun, that group will also guarantee that, over time, the population (local, nation, and world) will also continue to increase. So we'll see a population peak, a demographic swap paired with a population decline, and then a return to a growing population - likely with vastly different norms and values as a society than we have today.
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Of course my analysis is also flawed because it assumes no revolutionary change in the norms of today. These range from the unpleasant to consider such as changes to welfare systems, to the highly pleasant to consider such as the possibility of expanding our populations outward with the colonization of new planets and space in general. But ultimately, I do not see longterm population decline as probable in any scenario.
Your article also affirms this hypothesis. For instance in their prediction graph they show that Sweden will now, even with 0 migration, see consistent population growth. Sweden is simply playing out this scenario in fast forward due to a very small population paired with high migration rates which did little than accelerate the rate of demographic replacement. E.g. in the 90s Sweden's population was less than 9 million and with negative fertility rates, they were starting to see population declines. Then enter the migrations and now their population is greater than 10 million and their population is now back to growing, even without any more migration.
[1] - https://www.statista.com/statistics/241530/birth-rate-by-fam...
[2] - https://www.pewresearch.org/fact-tank/2017/04/06/why-muslims...