I agree the information typically provided by the private companies is limited. But they offer insight into health of the company.
The person reviewing these statements need to know how to analyze and reconcile financial statements. Also, as shareholder, you can ask their finance person to explain/elaborate certain items in the statements.
Shareholders' Equity section of Balance Sheet statement and Cash Flow from Financing Activities section of Cash Flow statement might also give additional insights. Company will be accounting for value of preferred shares, any dividend/interest paid on preferred on these statements.
I am not sure how much an employee can push without jeopardizing the job. IME, most owners and executives of such entities are typically egomaniacs who don't like being questioned unless someone has bigger stick than them. My personal experience is very limited and only as external investor.