In Harry Potter, wouldn't it have made more sense for Lily and James to make Dumbledore their secret-keeper?

James wanted one of his friends to be the secret-keeper, even though Dumbledore offered. James wanted Sirius to be their secret-keeper because they were very close, practically brothers. But, Sirius told James that Voldemort would be sure to target Sirius as their secret-keeper because that's what everyone would expect. So, instead of Sirius or Dumbledore being made secret-keeper, Lily and James made Peter Pettigrew their secret-keeper because no one would expect that. Obviously they were wrong to trust Peter and you know what happened and blah blah and yeah...I'm not sure if Dumbledore ever said why...maybe he did but idk