Part of declining birth rates is people having them when older. If you start at 21 you simply have more time to pop out 5-6 than if you start at 31 years old. As well, in your 20s you can physically handle lots of interrupted sleep and the like, in your 30s it's harder, and in your 40s it's awful (as I know, having had children at 40 and 44yo). I think the expense is largely just an excuse - children are hard work at any age, and as you get older you simply can't be bothered.
A government which wants to encourage a higher birth rate thus needs to encourage early parenthood. But that's not happening in any country where women are educated. As you increase women's education, birth rates decline - that's why even Saudi Arabia and Iran are close to 2 children per woman now.
The authors of Empty Planet contend the trend is happening everywhere, not just in wealthy countries, although you are correct to point out it is more obvious among them. Part of the reason why we don't notice it as much so far is that people are living longer. As such, median ages of populations throughout the world will rise, meaning that a relatively large population of older people will have to be supported by an increasingly smaller group of working aged people. Demands on social services to meet the needs of retirees will increase and as such presumably increase taxes on those who are working. Government programs to encourage larger family sizes cost money and, as the authors point out, are among the first to be cut when we have an economic downturn. As you perceptively point out, there is some upside to a world with fewer people. The environment will be a big winner and with increased urbanization it is possible that species driven to near extinction owing to the loss of territory to agriculture will make a comeback.
There are other things which will happen, too.
There will be social changes and civil conflicts in many countries. For example, Iran and Saudi Arabia have gone from 6-8 children per woman in 1980 to nearly 2 now. This means that at least 25% of families have accepted they won't have a son - this is a pretty enormous change, socially, as even the most patriarchal sexist father is likely to send his daughter to university if he has no son. Thus we will see an erosion of patriarchal systems - which is to say, the monarchical or theocratic rule - in those and similar countries. Those systems will fight back, if they can, or else simply collapse one day as the old Soviet Union did.
Urbanisation will at some point stop. In parts of the world like Syria and Egypt it's been driven by climate change - if your farm turns to a dust bowl, you eventually leave it. And thus climate change is a proximate cause of the revolutions and civil wars in those countries, since they suddenly had a million hungry unemployed young blokes in their cities. That change is largely irreversible in the near term, rain's not going to suddenly start falling in these new deserts.
But the other thing driving people off the land and into cities has been cheap energy. 12 men can plough an acre in one day, or one man and one horse (though the horse needs 4+ acres to sustain it) - but a tractor, as expensive as it is for rural peasants, can plough 20 acres in a day. Ammonium nitrate from natural gas likewise makes each area of land more productive, and pesticides and herbicides from oil do, too.
Fossil fuels are depleting, and though in principle elements like iron and boron can be endlessly recycled, fossil fuels once burned are gone forever. And so they will rise in price. Cheap fossil fuels give us cheap food, too, and drive people off the land and into cities, since the whole infrastructure of modern megacities can't exist without cheap fossil fuels. As fossil fuels become more expensive, the infrastructure of cities will begin to wear down and not be replaced, and they'll become worse places to live. Couple that with the rise in fuel, fertiliser, pesticide and herbicide prices, and we will see some people return to the land.
We won't go back to the Middle Ages with 10 of every 11 people involved in food production, but nor will be sustain today's Western systems with 0.5-2% of the population involved. I'm pegging something like around 1900, which was about 25% across much of the West - happening not tomorrow, but in a century or two.
At some point, economies will not grow any more. Economic growth ultimately comes from increased consumption of goods and services, but they can't be consumed without being produced, and that requires fossil fuels. And people aged 20-40 are the largest consumers, children and elderly consume much less. So there'll be neither the demand nor the supply.
Absent economic growth, the only way to FIRE will be a landlord lifestyle.
