A financial payment per child that depends on age of parent and goes down with age would be a strong incentive I'd say.
The elephant in the room doesn't seem to have been mentioned. Israel is (the only developed country) in a permanent state of war.
Could be, but South Korea and Taiwan are in similarly precarious situations and have abysmal fertility rates.
When you look at other countries which are constantly at, threatened by, or threatening war they also don't necessarily look great: Ukraine, Russia, Iran, China, North Korea, Finland, etc.
That's not to say this isn't a factor, but I think you'll need to add in enough extra details to differentiate from other countries that it will be very difficult to prove one way or another.
It seems to me Israel is clearly differentiated from these other countries.
South Korea and Taiwan have not mobilized and/or actually fought a real shooting war in the past 70 years.
Ukraine has 'only' been at war since 2014. It Russia has fought some very small border nations but never mobilized or fought anything approaching a major war until Ukraine. Similarly for China. It also seems to be the case that East-asian countries and former Soviet Union countries have depressed fertility rates for whatever reason.
North Korea seems like a case on to its own. Despite having nukes it be a stretch calling it a developed country.
Israel has fought major wars, insurgencies, and/or mobilized every decade or so. There are regular bombings and terrorist attacks. It also permanently occupies a neighbouring state that is similar in size and population. Whatever one thinks about Israel/Palestine and its politics I hope we can all agree that many Israeli feel a sense of being 'permanently under siege'. This seems reflected in the political landscape which was quite leftwing and progressive at its founding and has moved significantly to the right since.
It is of course also is very atypical in its cultural, religious, ethnic makeup, its geopolitical position and socioeconomic place and the large amount of migration so I agree with you one should be careful at drawing too many conclusions.
From Robin Hanson, via TheZvi:
Israel is the only developed country with a high TFR, and it's TFR has remained more or less stable at ~3 for the last 30 years or so. It's a good test case to see which of these factors might matter more or less than the others.
Israeli TFR is stratified by religious observance, with more religious people having more kids, but even secular Israelis are at replacement TFR - higher than the average for almost any other developed country:
So let's see how these trends apply to Israel:
More gender equality: gender norms in Israel outside the ultra orthodox community are pretty similar to what you might find in any other developed country. Traditional gender roles are maintained in the ultra orthodox community, but since many of the men study Torah all day, women are often the main bread winners, and expected to look after the kids on top of that.
Higher parenting effort: Anecdotally Israelis make for interesting parents. They are incredibly and obsessively protective of babies and toddlers (and continuously pester me as someone from England that my child too cold, or not drinking enough, or might choke on that food, or whatever), and mostly blasé about any kid above about 4 or 5, at which point they can go to the park by themselves, go shopping by themselves, and pretty much do whatever they like.
Long stiff career paths. If anything this is worse in Israel, because most people go to the army before university, adding an extra 2/3 years before they start making money.
Cap- vs cornerstone marry. There is no expectation in Israel to wait till you're financially secure before marrying. People will commonly marry (and have kids) in the army, in university, when just starting their careers. Doesn't matter, you'll sort it out together.
Grandparent less involved. Grandparents are very much involved in raising kids in Israel. They'll look after the kids when they're sick, pick them up from daycare if the parents are both working, etc. (something we with our parents in England miss out on). Whilst parents don't do much matchmaking, professional matchmakers are the standard in the ultra orthodox community, and not uncommon in the religious Zionist one. It's also normal for people to set up their friends.
More urban less rural. Israelis love in very dense cities, and the more religious the more dense. Ultra orthodox cities have some of the highest densities in the world.
Less fundamental religion. This clearly explains much of the stratification between different communities in Israel, with the more religious having (far) more kids.
Integrated world culture. This also helps explain the stratification, with the communities that are more insular, and take their norms from their own culture, having significantly more kids than those which are outwards looking.
Conclusion
Obviously this can't prove anything, but it does provide a case study in which factors might be important. My estimate is the most important factors (of these 8) in Israel is religion, integrated world culture, and cap Vs cornerstone marriage. Since changing the first two is unpalatable for most countries, the third sounds like a good place to focus. One way of doing this could be by providing strong tax incentives to marry, so that people marry earlier when they are most financially insecure, rather than waiting till they're well established financially.