The guards at the tomb would not have spread a rumor, because it would incur severe punishment.

While they were going, behold, some of the guard went into the city and told the chief priests all that had taken place. And when they had assembled with the elders and taken counsel, they gave a sufficient sum of money to the soldiers and said, “Tell people, ‘His disciples came by night and stole him away while we were asleep.’ And if this comes to the governor’s ears, we will satisfy him and keep you out of trouble.” So they took the money and did as they were directed. And this story has been spread among the Jews to this day. (Matthew 28:11–15, ESV)

What motivated the guards to spread the rumor?

It has been argued that the Roman guards would not have spread the rumor that they had fallen asleep while Jesus’ disciples stole the body, because the punishment for both offenses would have been severe. However, if the resurrection is true, the actions described in Matthew’s account are the actions we would expect from these soldiers, given the situation.

The tomb had been officially sealed. And the guards’ sole purpose for being at the tomb for three days was to make sure that the seal was not broken and that the body remained in the tomb. But now the body was gone, the seal was broken. They might keep silent, but there would have been no way to cover up their failure. If a report was to get back to Pilate, if he sent someone to investigate, there would be nobody to find. They could go to Pilate and give him a report, but either way, this would have been a clear case of the soldiers being negligent in their duties. It was not in the guards’ best interest to go to report to Pilate or simply be silent; they had failed in their duties and would have been at risk for a severe punishment or possibly death. This being the case, it makes sense that they would instead go to the chief priests to find an ally either to corroborate their story with Pilate or help them get out of it and conceal their failure. After all, the Jewish leaders clearly had a stake in the outcome of this event. On hearing their story, the chief priests and elders also did not want Pilate to get a report of this. They did not want this story to get out. This tale would provide fodder for the claims that Jesus was the Messiah, and “the last fraud [would] be worse than the first”. The Jewish leaders were not able to provide an unsealed tomb or a body, so they needed an explanation. In collaboration, they decided to promote the rumor that Jesus’ disciples stole the body.  Since only the guards and the Jewish leaders knew what happened and the three days had passed that the guards were to keep the tomb sealed, Pilate would have no reason to assume anything had happened, if both the guards and the Jewish leaders kept silent. The only concern would be if the rumor reached Pilate’s ears. Keep in mind, they did not ask the Roman guards to publicly announce this but to tell it to people in a way that would start a rumor, in a way that might not even reach Pilate’s ears. If it did reach Pilate’s ears, the Jewish leaders would spin another story that would validate the soldier’s integrity. The Roman guards didn’t have a lot of options, they could report to Pilate for certain punishment, or they could go along with the Jewish leaders demands and have them help protect them. They were also being paid a lot of money, which alone could have been enough incentive to spread a rumor, especially where the leaders have agreed to protect you. They may have simply seen the money as being worth the risk. The behavior of the Roman guards is not surprising if you take into account the context.