There is no evidence that the Betsy Ross flag promoted a sense of citizenship, self-sacrifice, or anything else during the Revolutionary War era. The story of Betsy Ross was first recorded in 1870, and that was from the memory of one of Ross's grandsons, who was himself only telling a story his grandmother had told him when he was a child. What is more, there was not, during most of the Revolution, a uniform design for a flag recognized by all of the states. Congress ordered that the flag should be the thirteen stars in a blue field with red and white stripes in 1777, but the design, while commonly seen in depictions of the war, was still not universally recognized.
However, Ross's story has become one of several American founding myths that have emphasized self-sacrifice or some other republican virtue, and indeed the proper role of women in the new country. Ross, after all, was a seamstress, a gender appropriate job for young women, and her story could be interpreted as evidence that women could best contribute to the nation by fulfilling their prescribed roles. The irony of this is that Ross actually owned her embroidery shop. She was far from a demure young seamstress, but a successful businesswoman. In any case, to the extent that symbols are important to American notions of belonging and nationhood, Betsy Ross is important for her mythologized connection to the nation's founding.