So far as I can see, Trump’s Executive Order today denouncing flag burning is a legal nothingburger.
Under the Supreme Court’s 1989 decision in Texas v. Johnson, famously joined by Justice Antonin Scalia, the First Amendment does not permit laws generally banning burning or other desecration of the American flag. Trump can’t himself nullify or overturn that precedent and isn’t promising to.
In common with other First Amendment decisions, Texas v. Johnson allows for certain exceptions, such as when someone desecrates a flag in the course of committing a different crime, or against the will of its owner, or does so in a manner likely to lead to an immediate breach of the peace. Trump makes noise about federal prosecutions or referrals for state prosecutions in those cases, but it’s not clear to me that many meritorious prosecutions were going unfiled.
In short, it’s a low-energy measure that pleases his base and shifts a bit of news attention onto an issue where he feels public opinion is with him. And look, it worked: I just spent twenty minutes writing on this instead of on what the Founders and Framers would have thought of his plans to use military and paramilitary forces to occupy American cities.