If I were a DA (District Attorney) looking out "stamp out" statutory rape, I would make the rounds of the hospitals, identify women who gave birth, or were impregnated when underaged, and go from there. But few, if any DAs, do this.
No, you wouldn't. First, you have neither the time nor the budget to do this. Second hospitals are not public property, you would need a warrant which the court won't give you because you don't have probable cause to believe a crime has or is being committed.
As a DA, I would of course, follow up on any complaints lodged by the victim, or even by her/his parents. And maybe I would prosecute a case where people were "caught in the act, or there were nude pictures, etc. May even an especially egregious PDA (public display of affection).
No, you wouldn't. You would follow up on complaints to the extent that you have the time, budget and manpower to do so and you would prosecute cases where you believe that you have a reasonable prospect of getting a conviction and where prosecution is, in your opinion, in the public interest. In the real world, that means where the police hand you a brief of evidence that is a lay-down misere and (for statutory rape) where there is a real power imbalance between the perpetrator and the victim - giving 2 willing 15 year old kids the label of 'sex offender' for life is probably not in the public interest.
But suppose there were no complaints or smoking guns. Could someone investigate based on something like a teenager "holding hands" with someone much older? To make the question objective, what has historically caused DAs to prosecute people outside of the above parameters?
With some exceptions, it is not the role of the DA to initiate or conduct investigations of criminal actions - that is the role of law enforcement officers, usually for this particular crime, police officers.
Notwithstanding, the general rule in western, liberal democracies (which the United States, with a certain generosity of spirit, can still be considered) is that citizens are allowed to get on with their lives without day-to-day interference from the state. That is, law enforcement officers do not 'go fishing' for crimes, they investigate crimes that they have a probable cause to believe have actually happened, either because they themselves saw it happen or someone has reported it to them as having happened. The term for having a law enforcement officer following you around waiting for you to break the law is 'harassment' and may itself be illegal.
A law enforcement officer can initiate an investigation based on anything but as no law enforcement agency has unlimited resources, they tend to follow only those investigations that might lead somewhere.
For example, it has been known for parents to hold the hands of their children and even to publicly display affection towards them and parents are often "much older" then their children - this would not generally be grounds for initiating a statutory rape investigation.