I can see a reason why receivers of cash tips would not be honest with how much they earn, especially to the government.
1. The BLS knows this, so they get their data from multiple services--not just government agencies.
2. The IRS knows this, so are more likely to audit people who work in industries that receive tip income.
3. If the BLS numbers were vastly underestimating bartender income because of unreported cash tips, you would expect bartender income to have risen much faster than other similar low wage jobs over the last 20 years as cash has become less common. This hasn't happened.
I find it highly unlikely that a server is making anywhere near 60% or so of their tips in cash these days.
Maybe there are a few locations with that high of a mix of cash, but not enough for that to be anywhere near normal.