Several things here:
1) It would have been a far greater risk to have him stay, and risk getting questioned/observed by police.
2) The prospect of going to jail is utterly terrifying to a 9-year-old. That would be more than ample motivation for him to NEVER admit his wrongdoing to ANYONE, even as a 9-year old child.
3) In fact, new interview footage teased last night shows BR being interviewed by cops. When asked if he had any secrets, he tells the cop "I'm not telling you."
4) If BR ever blabbed at the Whites (not likely, IMO, for the reasons stated above), the Ramseys are still in a win-win situation. Let's say he eventually cracks.... his parents are guilty of a cover-up, they are not likely charged due to the enormous sympathy of parents simply being motivated by "not losing their other child." BR never cracks....they all get away with it and the Ramseys don't lose their son. By adopting a cover-up for their son, it's a low-risk, high-reward strategy. The other option is having BR confess, and the end result is they gain nothing and not only lose their other child (they probably didn't know at the time that BR could not have been tried for murder), but intense media scrutiny focuses on a highly dysfunctional family. Therefore, a cover-up for their son is the most rational course of action.
Well the GJ voted to indict, and we know they couldn't have been voting to indict Burke. So it appears to be a very high risk strategy that might possibly put them in prison. Once they know BR is safe from prosecution their lowest risk strategy is to fess up to staging to "protect" Burke.