Why did the President still send the FBI after Benjamin Gates?

Question

In the film, National Treasure: Book of Secrets, Benjamin Gates must locate the Presidential Book of Secrets. The only person who knows the location of the book is the President of the United States. Benjamin decides to attend a birthday party for the President at Mount Vernon. Benjamin then intrigues the President with the history of Mount Vernon and lures him to an underground tunnel to get him alone to ask about the location of the book. He has the President alone and trapped in a room. Ben explains everything to him and the President helps him and gives him the information that he needed:

President: The following conversation never happened...The book exists.

Gates: Where is it?

President: Where else do you keep a book? The Library of Congress.

The President then goes on to tell Benjamin that he can't tell anyone why Benjamin held him against his will:

President: Gates...Two Hundred people will know that you held me against my will. I can't tell them why. Unless you find what you're looking for, you will be charged with kidnapping the President.

The FBI is then chasing after Benjamin and his associates throughout the remainder of the film. After Benjamin found what he was looking for, he is brought into custody by the FBI. The President is waiting for Benjamin when he arrives and tells everyone that he wasn't kidnapped and that Benjamin saved his life:

President: As I recall, we were exploring a hidden tunnel, and a door closed accidentally and this man saved my life.

Why didn't the President just tell everyone this story to begin with? What was the purpose of the President saying that he was kidnapped? Is there some out-of-universe explanation that filmmakers chose to incorporate into the film regarding the kidnapping of the President of the United States?

Answer 1

What was the purpose of the President saying that he was kidnapped?

Obviously, he didn't tell that he was kidnapped, but since President was locked inside the tunnel, everybody assumed that he was kidnapped. Since this is a matter of safety of president, FBI and police started to hunt Benjamin down. All those chasing was done because of that assumption.

Remember that FBI and police already left when news about kidnapping of President spread like wildfire.

Why didn't the President just tell everyone this story to begin with?

He could have told this, but nobody would have believed story about that treasure. Therefore it was necessary to find that treasure, and give it to government.

Another motive of president behind this could be that FBI and police chase might make Benjamin to find that treasure anyhow to get rid of those charges. And it worked out in the end. All the charges on Benjamin and his friend were removed and they were rewarded for this.

Also he revealed that secret book which was, of course, a rule violation.

Is there some out-of-universe explanation that filmmakers chose to incorporate into the film regarding the kidnapping of the President of the United States?

This could have been an attempt to make the movie more thriller. Though I could't find any official answer/sources regarding this.

One source

Question: Why does Ben need to find the city of gold to beat the kidnap charge? The President's story of "We got trapped and he saved my life" would work just as well without the discovery.

Answer: I've wondered that myself. It is a rather obvious plot hole, but it appears that the writers thought it added to the story's "suspense" by having Ben eluding the federal authorities while racing to find the treasure

Answer 2

I always thought the movie required quite a bit of suspension of disbelief. As far as I know, there's neither an in-universe nor out-of-universe explanation as to why the President would cover for Gates after the City of Gold was found. To begin with, nobody ever had to know the President was held against his will, so that whole part of the dialogue was far-fetched. It's just one of those things you have to accept and not put too much thought into, much like Indiana Jones' ability to know how to circumnavigate the traps at the beginning of the first movie.