The point you are labouring here, that the 'authorities' at the time could have any scotched rumours of the resurrection of Jesus by simply producing the body, assumes that; a) the 'authorities' saw Jesus as being a special case, and b) they knew of the rumours of Jesus being resurrected in the days immediately following the alleged execution.
My last response on this was a dismissive one liner, so let me expand a bit on the point.
Let us, for the moment, assume that the Christian account of the early church is true. If the authorities were concerned about its rise, why didn't they produce the body? In fact, there seems to be no record at all of them doing anything to counter the rumours of Jesus' resurrection. Christians say that this is because the rumours were true and there was no body. However there is a better explanation.
The obvious answer is that the authorities were not aware of any rumours to counter. Either that,or they didn't see Christianity as a threat. It's not as if it was the only Jewish Messianic splinter sect or the only Greek mystery religion. The Romans were quite tolerant of different religions and there were many different beliefs in the empire.
Another possibility is that the authorities did produce the body but, for some reason, the documentary evidence hasn't reached the present day. For a long period in European history, Christians were the sole custodians of the preservation of written documents. Why would they lose an account of a failed attempt to discredit Christianity. The answer, of course, is that they wouldn't, but they would "lose" an account of a successful attempt.
The mythicists also have an answer to the problem. If Jesus was a mythical god-man, only morphing into the current legend at a later date, the evidence would be exactly as it is. The authorities wouldn't have countered Jesus' resurrection because everybody knew - even the proto-Christians of the time - that Jesus wasn't a real human being.
Of all the possible explanation for the silence of the authorities, the Christian one is the least credible.