If we apply the same thinking and logic to the "redress of grievances clause" there is nothing in the constition that prohibits the president from granting a redress of grievances under the first amendment. Indeed it is as much an implied power as the Supreme Court's implied power to rule on the constitutionality of acts of congress. Isn't it?
Using the same reasoning behind Marbury, the president could easily and rightly claim this power. Indeed, IMO, the president has always had this power. It is simply that it has never been done before. Perhaps now is the time to test that power and authority. This is as good a case as any.
At that point in time, the governmental body to whom I bring my grievance has an obligation to act according to their office. While the wheels of justice grind slowly in the legislative and the judicial, the whole point of an executive is that he can order an immediate remedy.
The word "redress" does not mean "final solution" as I understand it. It leans more in the direction of: "a process for remedy."
Therefore, the immediate granting of a process of remedy is in the hands of whatever branch is approached. The final solution would be in the hands of "government."