You don’t get to make up your own rules. Just because you think a court should do the investigating, doesn’t create the obligation, or the rational expectation, that one will.
You have to present a case to the court. You don’t go to court and tell the judge you think there’s a problem, but you don’t have any evidence unless the court investigates.
The case was made when the secretaries of state, the parties, and the congress admitted they didn’t check the guys birth certificate. No one is asking the court to make the case. They are asking the court to uphold the constitution and oath taken by every judge. Judges grant discovery all the time. The have to compel the guy to present the records because he refuses to do so. The court should demand the “best evidence” be presented so that the issues can be decided on that basis.