I mean - isn't the birth of every person in America documented with birth certificates, and so forth? Wouldn't it be pretty simple for a police investigator to check when somebody was born?
Npt necessarily.
People born through "the system", that is, in hospitals and whatnot, are usually documented. But there are still lots of people born outside the system, people in very remote or rural areas, people born at home using a midwife rather than in a hopsital, and so forth. Those births aren't necessarily recorded in any official capacity, and when they are recorded they're generally recorded based on the testimony of people close to the family. For instance, the parents go to the records office and attest that the child is theirs and that the birth occurred on Day X. If they're lying to protect their cult, who would know?
The local authorities might be in on it, too. There are some polygamist sects that are incorporated into small towns, and the sect leaders are the legitimate government authority by virtue of being the mayor, sheriff, town councilmen, official records-keeper, or whatever. They could be in a position to falsify the official records or do other officialdom-type things to undermine the prosecution's case.