Felony crimes seems like a fair filter.
The sad thing is most politicians ARE criminals and folks who seek high-gain / low effort jobs.
Filtering out those who have unpaid parking tickets, tax issues and misdemeanor offenses would leave very few worthy ones left in the lottery pool. In fact, having tax problems could be a special filter and give us some real folks who understand revisions are needed in the Federal bureaucracy. Maybe only folks with a history of tax liens could be put in charge of the IRS.