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.