If it was made universal I think the cost of living would rise so much that it stops helping.

In my ideal world, cost of living would be matched to the current needs of replacing babies.

So say for example the program is a success and now everyone has large families with children. That's when the government can divert resources from fertility and use it to fund other projects like science research, healthcare etc.

As a result, the incentive for birth rates will decline and people can focus on having meaningful careers again.

Otherwise, the only solution I can think of for having excess children is to have them colonize other lands. It's sound sci-fi, but sending them to colonize Mars would solve this issue.