I never heard that about their leaders. Have any examples or documentation about that?
Not really. Just various things I've read over the years about them. They were in more or less continuous existence for almost a thousand years. For most of this time, they had approximate equivalents on the Christian side, the Knights of Malta or Rhodes, various Venetian and Genoese groups. Both sides used POWs as galley slaves.
The difference is that the "Christian" countries moved past piracy and the corsairs stuck with it well into the 19th century, until eventually putting up with them became more of a hassle than going in and squashing them, which the French finally handled, starting in the 1830s if I remember the dates correctly.
There are obvious parallels to today's terrorists, who mistake our unwillingness to use all the force available to us for weakness. Of course, the corsairs never had a chance of getting hold of a nuke!