I like your example of the Unabomber’s brother, whose fraternal betrayal was justified and yet accompanied with the sense of moral burden that reflects good character. And I share your sense that our “tell-all, sell-all” culture makes for a public life inhospitable to loyalty.
It leads me to wonder whether it is quite the case that the celebrity/therapeutic culture breeds “loyalty to oneself.” When Monica Lewinsky reveals all to Barbara Walters, isn’t there a kind of self-betrayal? Betrayal is not only a way of (mis)treating others; it can also be perpetrated against oneself. We have spoken of loyalty to friends and patriotism to a people. The parallel virtue is integrity, a kind of solidarity with oneself.
We sometimes think of integrity as wholly a matter of honesty, or truthfulness. But integrity is a more complicated virtue, having something to do with weaving together the strands of one’s life in an integrated whole. This is why it makes sense to think of integrity as a kind of solidarity. Blurting out one’s innermost thoughts and feelings in inappropriate circumstances may be honest and truthful, but it is also a kind of “snitching” on oneself, a kind of betrayal.
We admire Joe DiMaggio not only for his greatness as a ballplayer but also for his integrity as a person and as a public figure. His integrity did not consist in honest or truthful revelations, but in precisely the opposite–in discretion and restraint. As you point out, he even managed to protect the privacy of his marriage to Marilyn Monroe.