Actually, it makes perfect sense.
Flags are lowered to half staff when the entity represented by the flag is in mourning. In other words, state and national flags would be lowered to show that the state and nation are in mourning. If a city flag was being flown, that would make sense to lower to half staff, too.
Flags that are not the official symbol of a governmental entity are not lowered to half staff.
For example, a signal flag would not be lowered to half staff. A corporate flag could be ordered to half staff, although it would generally only be done for an employee of the corporation.
The flag in question is the Confederate Battle Flag (it is not the Confederate Flag, despite the common error in labeling it as such). It doesn't represent a governmental entity at all. It represents an army that no longer exists. There is no particular reason it ought to be lowered.
I learned all of this shortly after the death of Princess Diana. There was outrage in Britain that the royal flag atop Buckingham Palace was not lowered to half staff. I read an article by one the heralds there bemoaning the lack of knowledge on the part of the press that was printing articles questioning why the flag was not lowered. The royal flag is apparently a signal flag. It is flown above the royal residence if and only if The Queen is present at the residence at the time. Only the Union Flag (which he pointed out was not the Union Jack) should be lowered during times of national mourning, and that the Union Flag was being appropriately lowered.
On the other hand, The Queen and/or her staff showed more sense than the South Carolina government, apparently. Despite the Herald's objections, they decided it was better to look good in the public eye than to follow ancient, but widely misunderstood, traditions related to flags, so they lowered the royal flag to half staff.
I would be curious to know if there are flags flying at South Carolina National Guard bases, and if those flags have been lowered. I'm sure that the state flags probably have been lowered, but I'm curious if the National Guard of South Carolina has its own flag, and whether it has been lowered.