Which system do you have in mind and what's going to fall apart 'without the lies'?
I'll give you one example. France has recently made it illegal to do paternity testing. Someone clearly understands that without all the lies their population of single mothers would skyrocket.
If by "recently" you mean 5 years ago. And by "illegal to do paternity testing" you mean "illegal to do secret paternity testing by private citizens without a court order". So your claim seems very false, and doesn't back up your idea that "their populations of single mothers would skyrocket".
To be clear: France has made private paternity testing illegal, while you can still do it through a court decision.
Still weird and the reasoning behind it seems backwards.
But why make private testing illegal. It's very clear that only a very small percentage of people who wish to get a test done would approach the court, and how many of them would get the order? E.g. I suspected my child was not mine, so I got a test done secretly only to find out my suspicion was wrong and I was actually the father. Would I have approached the court for the test, if it was illegal? Most likely not, and I would have lived the rest of the life in a doubt. And even if I did approach the court, it would have affected my already strained relationship with my wife.
You'll be even more surprised when you hear that they said something alo g "this rule is good for the piece of mind of the French families".