Yes Walt. It does. Article I, Section 1 states "All legislative Powers herein granted," meaning in this article, "shall be vested in a Congress of the United States"
One of the powers granted in article I is that of suspending habeas corpus and section 1 states that it shall be vested in the Congress.
Now, tell me where the Constitution forbids the president from suspending the Writ?
On a strictly technical reading, in Article I, Section 1 because it specifically vests that power in the Congress and not the president.
The Writ has nothing to legislative power. It's a privilege.
I am still waiting for you to tell where the Constitution -specifically- prohibits the president from suspending the Writ.
Walt