Your topic asks Why didn't Jewish leaders dig up the dead body of "Jewish"... did you mean to say dig up the dead body of "Jesus"?
Because urban legend is a Big Universe.
Maybe because by the time Christians started developing traction, several years had passed. And Jesus had been buried anonymously as a criminal.
Or, maybe Jewish leaders argued among themselves whether it was better to ignore Christians or actively suppressing them including by murder. And maybe they went back and forth on this.
I'm not sure about the urban legend part because some scholars say that the belief in the death and resurrection of Jesus began as an oral creed that even predates Paul's writings. In fact, some scholars date this creed to even a few months or years after Jesus's death.
Here's from a skeptic and historical scholar, Dr. Richard Carrier responding to a Christian claim:
I keep hearing Christian apologists insisting the Corinthian Creed (1 Cor. 15:3-8) can be reliably dated to the 30s A.D., just years or even months after Jesus died. Can you direct me to a solid refutation of that claim?
The answer is no. Because there is no refutation of this claim—other than “maybe possibly it originated later,” which is the logical fallacy of
possibiliter ergo probabiliter (“it’s possible, therefore it’s probable,” see
Proving History, index). In fact the evidence for this creed dating to the very origin of the religion is amply strong; and there is no reasonable basis for claiming otherwise.
Your other two explanations are more plausible.