Oh yes they will. Remember Public Enemies? President Luthor is back in canon so I assume the events of his fall roughly are as well. And Clark fought the military as part of Johns Secret Origin. I may miss the New 52 origin but current Reborn Clark certainly isn’t a boot licker who will bend over for anyone with an ounce of “authority”.
Honestly though why do you want to see a “realistic” fallout? We saw that with Truth already. Clark got fired and everyone hated him. No real point rehashing that when we’re already redoing the “Superman loses his SI” plot. Let’s see something different if we have to go down this road again.