I think you might be confusing two different search authorizations; people make this mistake all the time.
The first common authorization for a search comes from Terry, and allows the police to conduct a cursory search of the outer layers of your clothing (not including pockets) to attempt to detect weapons. The grounding for Terry searches is exclusively officer safety. Terry searches are the basis of "stop and frisk".
The second common authorization is "incident to arrest". The police can conduct a thorough search of your person, effects, and immediate vicinity if they actually place you under arrest. The grounding for this search is both officer safety and collection of evidence. In fact, I think even Rayiner is a little wrong here; the incident to arrest authorization doesn't merely concern destruction of evidence, but is specifically concerned with collecting all the evidence that could be the fruit of an arrest, whether or not it's at risk of destruction. The meaning of search incident to arrest goes all the way back through Common Law.
The Obama DoJ is clearly referring to search incident to arrest.