The US economy won`t collapse, if there aren`t any gold stored in Fort Knox.
According to Wikipedia:
"As of November 2017, Fort Knox holdings are 4,582 metric tons (147.3 million oz. troy), with a market value of over $100 billion. In contrast, the GDP of the United States was $19.4 trillion as of April 2017."
This year,the US GDP is more than 20.413 trillion USD.
I don`t think that bitcoin popularity will increase ,if the US government has no gold reserves.
Nobody cares about that.People are just ignorant.
It is all about Faith/trust in the government. If there are no Gold in the vault, then the US government will lose a lot of credibility and the US citizens and the rest of the world that are using the US Dollar would lose faith in them. That little piece of paper, called the US Dollar is 100% built on the faith that people has, that the government would guarantee it's value to the bearer.
The Gold's
value has little impact on what the damage would be, if it was found that there were no Gold in Fort Knox and the government knows that.

The US government confirming that there is no gold in Fort Knox won't be the end of the US Dollar as the they won't obviously lose credibility on one thing, in fact they could probably have a good excuse when they finally decide to tell something straight publiy like that they where only using Fort Knox as a decoy place for them to protect the real gold the US is owning somewhere is a undisclosed location. Also there are several occasions where the government has been found hiding the truth and the most recent ones are about their war in the middle east but did they lost their credibility during that time? Of course not being the most powerful country in the world it is hard to bring it down.