You are wrong. Read the Constitution as well as the decisions of the court. What part of "The privilege of the Writ of Habeas Corpus shall not be suspended, unless when in Cases of Rebellion or Invasion the public Safety may require it" are you having trouble with? It's part of the section on Congress.

You seem to forget that Congress wrote the writ into law. Again, from Ex Parte Bollman:

"Congress has expressly given this power to this court, by the 14th section of the act of 24th September, 1789, commonly called the judiciary act. This section, according to its true grammatical construction, and its apparent intent, contains two distinct provisions. The first relates to writs of scire facias and habeas corpus; the second to such other writs as the court might find necessary for the exercise of their jurisdiction. As to writs of scire facias and habeas corpus, which are of the most frequent and the most beneficial use, congress seems to have thought proper to make a specific and positive provision. It was clearly and obviously necessary that such writs should be issued, not merely to aid the court in the exercise of its ordinary jurisdiction, but for the general purposes of justice and protection. The authority, therefore, to issue these writs, is positive and absolute; and not dependent on the consideration whether they might be necessary for the ordinary jurisdiction of the courts. To render them dependent on that consideration, would have been to deprive the courts of many of the most beneficial and important powers which such courts usually possess."

And only Congress can do it.

"If the question, respecting the power of this court, under the constitution and the act of congress, if not [8 U.S. 75, 87] under the common law, to issue the writ of habeas corpus ad subjiciendum, were still open, it ought, on these principles and authorities, to be decided in our favour. But it is not open. It has been twice solemnly adjudged in this court. First in the case of Hamilton, 3 Dallas, 17. not long after the court was organized; and very recently in the case of Burford. (Ante, vol. 3. p. 448.) We contend that the case is settled by these decisions, and that it is no longer a question whether this court has the power which it is now called upon to exercise."

The existence of the power to suspend the writ was manifestly declared by the founders to reside only in Congress:

Article I, Section 9:

"The privilege of the Writ of Habeas Corpus shall not be suspended, unless when in Cases of Rebellion or Invasion the public Safety may require it."

They expressly enumerated the power to suspend the writ as among the Legislature's powers:

Article I, Section 9:

"The privilege of the Writ of Habeas Corpus shall not be suspended, unless when in Cases of Rebellion or Invasion the public Safety may require it."

You are forgetting that the Constitution limits the power of government. If a right is not granted to the government by the Constitution, then they don't have it. Nowhere do we find the President being granted a right to do anything about habeas corpus. Only Congress can suspend it. It is part and parcel of "judicial power."

By placing a limitation on its use only in one place in the Constitution, the founders expressly denied the power in other places. While the Constitution is not a laundry list as it applies to the people (the Ninth Amendment still stands), it is as it applies to the government. Since the President is not declared to have any power over habeas corpus, that necessarily means he is denied it.

Only Congress.

You might want to read the jurisprudence on the subject.