[BitBucket]

"In any case, the way a suspension of heabeas corpus would work in practice is that the executive detains someone, the court is petitioned to require the executive to produce the accused and its evidence, and the executive ignores the summons, pleading that because of public emergency it is released from this obligation."

Correct. But the right to do that is granted to Congress, not to the President. It wouldn't make much sense to grant the President the right to suspend judicial oversight of himself.

[Non-Sequitur]

But the right to do that is granted to Congress, not to the President.

The Constitution does not specify that the power to suspend is granted exclusively to Congress.