Not reporting income has nothing to do with wanting to keep firearms transaction between individuals private, that is not reporting to the govt.
For one thing, background checks do not prevent crime and some reports state there have nearly zero prosecutions for failed background checks. So the entire premise of background checks somehow making anyone safer is a fallacy. Criminals usually steal guns or trade for guns amongst themselves so they would never be affected by background checks. This was true even before background checks were mandated for FFLs. Do you really think a criminal wants to pay full retail for a gun he is likely to throw away after his next use?
Secondly, background checks cost money. They are run through your state's law enforcement branch and cost at least $10 each time. Not a huge amount but it is $10 flushed down the toilet for no good purpose.
Background checks are a burden to the law-abiding, have no effect on safety, and create a tempting data source for govt to abuse its role and retain that data in lieu of an outright registry.
"The ultimate authority ... resides in the people alone. ... The advantage of being armed, which the Americans possess over the people of almost every other nation ... forms a barrier against the enterprises of ambition."
- James Madison