I've never thought much about it other than the president can order it at half staff when there is a national tragedy. I didn't realize there was much controversy around it until you posted and I checked.
Apparently, by U.S. Code it must be flown at half staff for 30 days after a president dies and 10 days after a vice president, chief justice or speaker of the house dies. More days are being added beyond Memorial Day, such as Bush adding September 11.
Obama ordered more half staff days than any president before him and some people thought it might cause it to lose its meaning. Should you lower it when civilians lose their lives to mass shootings or members of the military lose their lives in service? After the Orlando shooting, when Obama ordered the flags at half staff, an Alabama county administrator got press by refusing:
"When the flag is at half-staff, our country’s head is figuratively held low, and quite frankly, I am not willing to hang my head down because of a terrorist attack against our people and our allies,” he said. ” I am not willing to hang my head down because evil shoots up a church, school, or movie theater. We need more than a gesture as a response."