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.

That's why it is in their best interest to keep up the lie if there's nothing in those vaults? They want to keep a semblance of credibility in the face of their recent loss of the people's trust. It also helps to keep the price of gold down because that's an extra several million ounces in supposed circulation, which also drives up the dollar's value.
I don't think any government would ever let their people know that there is an empty vault and several million ounces are missing anyway. They'd be shooting themselves in the foot.
As far as bitcoin is concerned, I don't think that it would encourage people to invest in it if they found out this gold is missing. If anything, it'll fuel the conspiracies of the goldbugs and they'd rush to buy more because they'd see the value as higher than it actually is.