Why the US Navy relied on rear admirals for senior command

Did the US Navy have senior grades above rear admiral during the Great White Fleet era?

Generally no, apart from Dewey's exceptional status. The navy considered rear admiral sufficient and saw the structure as helping keep senior officers younger.

Apart from George Dewey's exceptional rank, the US Navy saw little need for grades above rear admiral. Rear admiral was considered sufficient for senior command. The limited hierarchy also helped keep the senior officer corps comparatively young.