One response to this quote would be a reminder that for well over a century Confederate soldiers were not allowed to be buried inside a national cemetery. There have been exceptions, such as the alcove created for Confederate soldiers at Arlington National Cemetery. Usually Federal park policy was like that in place at Gettysburg in which officials would not take chances that a body found on the battlefield might be that of a Confederate. Unless a set of remains could be confirmed to U.S., they could not be interred anywhere within the bounds of the park.
The reason for this policy is encapsulated within your Sheridan quote.