If you did this you risk women being 'exposed'. In many cases this potentially puts their life at risk.
Yes their child would know their parentage, but at what price? Is it in their interest to always know the truth? It is debatable in many cases.
On the other hand blokes could be tracked down more easily for maintenance. There is also the potential to track criminals more easily.
But that really is the can of worms. As it stands the police can only keep your DNA on file if you are convicted.
Do we want a huge database of our dna out there? What could it be used for? The assumption that it would be for society's benefit is a few Hollywood movies short of the probable realities. The potential for beaucratic kafkaesque nightmares is very real. If someone has your dna on file, you can alter the file to claim someone else was there. The information this data holds is invaluable - its a hackers paradise. You run the risk of state sponsored attacks and the development of weapons to target either individuals or groups. Or other espionage type issues.
Then theres the commercial side of this. How much is the information worth to insurers? Would this limit your life opportunities? Or next frontier is indeed those with certain dna qualities perhaps being favoured for sports or certain jobs. Like being born poor, being born with the wrong dna will be a handicap - perhaps greater than race. Then there the idea of 'precrime' which minority report touched on - with some people maybe being deemed to be predesposed to commit crime - but only to a certain degree of accuracy.
And the medical ones. Would you want to know that you had a 80% chance of not living part 40. And if you knew that, what would it do to your attitude to life? Would it encourage people to go on mass shootings as revenge as they approached their life expectancy. Or lead to people never bothering to try to seek a career as it 'pointless'. Would you really want to know that you are likely to die of dementia or a very painful disease?
Think of potential of genocide on steriods with this.
Pandoras box may well be already open and this is all an inevitability we are on the road to.
My point is you need to think WAY beyond the idea that it will help paternity issues in a good way. It also has a remarkably dark and deeply unsettling set of issues that it could unlock.