Doesn't the flag symbolize my freedom to burn it if I want to?

There is no historical precedent and scant legal precedent for such a bold and sweeping assertion. The flag is the symbol of our nation and it symbolizes what the people say it symbolizes, and the vast majority certainly don't think that includes the freedom to desecrate it. You will have a hard time convincing the average American that the right to desecrate the flag is a right deeply rooted in the First Amendment.