Go ahead and you will cause a massive uproar with untold damage to society when fathers will figure out that a lot of babies are not theirs. One of the reason why private paternity tests are illegal in France.
Interestingly if sufficient information is stored it would solve the problem. Cost of children would be directly pointed at the biological father. Thus not enslaving the cheated man for massive cost. Vulnerable must be protected from those with power of the system behind them.