From the framers:
Men are endowed by their creator with certain inalienable rights, and to protect those rights, mankind has formed Government, which has power because it is given that power by its citizens. The president is the chief of the executive branch of the government that is there to protect the rights of the citizens.
Every law that the Government passes has to respect and safeguard the rights of the individual, whether that individual is the President or the guy standing there protesting the President.
The man on the corner who is protesting the President is exercising his right to free speech. If he happens to be wearing a sidearm while doing so, he is simultaneously exercising his right to bear arms. As long as he threatens no one, he is not violating anyone's rights, and to molest him is to violate his rights of both speech and arms.
Yes, the SS has the duty to protect the President, but nowhere in the COTUS are they granted the power to trample the rights of the citizens in the name of protecting the man who is SUPPOSED to be making sure that those very rights are protected, nor does the COTUS grant power to Congress to delegate that power to the SS.
The President is both a man and a symbol. The man has rights, the symbol has powers. The man has no more, nor fewer, rights than any of us. The symbol has power, and those are specifically laid out in the COTUS.