It didn’t “obviously” fall to the President, as the President has no constitutional authority to suspend habeas corpus—that is textually committed to Congress. It boggles the mind to say that a power textually committed to one branch of government “obviously” falls to another if one branch of government chooses not to exercise that power. That’s like saying that the President can make laws when Congress is not in session. It’s just wrong.
Indeed, as I noted above, none other than the Chief Justice of the United States Supreme Court held that Lincoln’s actions were unconstitutional—a ruling that Lincoln ignored, incidentally.
Congress disagreed with your assessment about President Lincoln’s power to duspend habeas corpus, and so do I.
Under the Militia Act of 1793, it did fall under the authority of the president when Congress was not in session.
Did the confederacy have such an act that allowed Jeff Davis to declare martial law in many areas of the south?