I think in a highly financialised free market like the US the executive is pretty powerless to really hide important economic data like this, even within their own country.

There would be so much alpha in knowing stuff like the true employment rate that private agencies will be extremely well-funded to collect this data (in fact this probably already happens for some data, there might be alpha in having a second opinion even if you think the government data is trustworthy).

I think the worst-case outcome would be that the mass populace doesn't have access to the info because it's paywalled. But to the extent that journalism continues to exist, journalists will know the employment rate, GDP, that kinda stuff.