To some degree, yes, it is unifying. But there are elements of the national anthem itself which are quite political.
1. It is a war anthem, emerging out of the War of 1812.
2. A later verse of the poem is about slavery.
So while playing the anthem may be a popular political act, it is still a political act. Besides, it was hardly ever played before or during sporting events until World War II. The first time it was played was in 1918, in the midst of US involvement in World War I. Sometimes, such acts can permanently change the narrative. For example, how often was God Bless America played during seventh inning stretches before 9/11? But you can't take it away now, even though 9/11 is more than 16 years ago. Taking it away would be criticized as unpatriotic.