Shkreli got busted for fraud, not fiduciary duty. He paid our investors in one investment with profits from another.

Unlike Elon Musk he didn’t do so publicly and with shareholder approval.

Regardless of that execrable Shkreli, you have de jure fiduciary duties if you are on the board of a private company, yes. At least in the most common forms of incorporation. I am not even sure if it requires you have other shareholders, let alone if you are a controlling shareholder or they are accredited investors.