Fun fact, in 2005 they tried to do this through legislation. It was bipartisan. It was so bipartisan that they only needed 1 Republican sponsor, and 4 Democratic cosponsors were more than happy to sign on. This was made with the express purpose of taking it to SCOTUS to overturn the previous ruling that made flag burning as protest legal.

As a fun exercise, try to guess who the first Democratic cosponsor was.

If you guessed Hillary Clinton, you were correct!

And yes, 2005 was far too late in the Bush years for Democratic senators to be pulling this shit.