Would this encourage more couples to have children?
Babies are not like buying clothing, that you try it on for a few days and return if you don't like. Primary reason most people don't have more kids is the economic stress and affordability. Of course there are people who genuinely do not want children, no matter how much they earn. Those are the minority though.
And yes, not all countries face all problems listed above, it is just a collection of problems that countries face at different levels. But in general, social safety nets are disappearing everywhere. We can't expect responsible people to have more kids in this situation. Of course irresponsible/gullible people will still have kids even if they can't afford, which will only add to the problems
The data is robust that some don’t want children out of economic reasons, and others don’t want them out of lifestyle choices (prioritizing self over a thankless job). Across several national pro natalist policy programs, the evidence shows that even when enormous amounts of benefits are provided, it barely moves the realized fertility outcome.
(40% of pregnancies in the US and internationally, annually, are unintentional, and we have enough humans we don’t take care of already [1], we should be radically empowering as many as people who don’t want to have kids to not have them)
This sounds just like some people approach pets
Consider: the government pays a salary to each married family while they raise children; the salary would be equivalent to a blue-collar job, and it would scale with the number of children up to a point (e.g. 4 kids).
I strongly believe that'd lead to many more marriages and childbirths. Many people not interested in raising kids would prefer it over a "regular" job. Families with adoptive children would also be paid, so it could decrease adoption difficulty and stigma as a side-effect.
However, some people will game this policy, and it would be very expensive to implement.
I believe most of countries have orphanages already -- and what you're suggesting already exists in some countries (I do believe we still have that in Brazil).
While that could increase the number of people, orphanages are not great places to raise a child (with rare exceptions). Imagine you growing up with a large group of other child, and nobody actually take the time to take care of you. What kind of person would you be today?
Finding people willing to adopt newborns is also fairly easy now.
Yet nobody cares in the west to implement even this. Having children became a luxury.
This is a problem not solved by a startup.