No, that isn't the alternative. The alternative is having an incentive for having a sufficient number of children to prevent population decline, combined with separate subsidies for child support (professional childcare, etc.). Even better, drop the emphasis on mothers: allow either parent to claim the tax deduction, or devise a structure in the deduction is graduated with a return to the workforce.
There are plenty of reasonable ways to incentivize people to have children, nearly all of which are applied elsewhere in Europe while also helping (if not outright avoiding) attrition in the workforce. Hungary's approach represents the significant deviation here.
This is a pretty ridiculous distortion: not only is childcare not "outsourcing" of one's child's development, but it's also one of the oldest and most traditional vocations. It predates our modern economic structures by thousands of years.
There are plenty of reasonable ways to incentivize people to have children, nearly all of which are applied elsewhere in Europe while also helping (if not outright avoiding) attrition in the workforce. Hungary's approach represents the significant deviation here.