Switch to online statements pleez pleez pleez

This gives me a sense for how much a financial institution wants people to switch away from paper statements:

Win A Windfall With Online Statements
$10,000 Grand Prize, Plus 50 $1,000 Winners
Switch to online-only statements by November 16, and you'll be automatically entered for a chance to win in our sweepstakes.

I like the idea of online statements and bills because I hate filing and don't like the waste of paper. But for important stuff like utilities and financial records, I'm continuing to insist on paper. My reason is that if something happened to me so that I couldn't monitor them myself, then someone else would have to know and keep track of all the accounts and all the passwords for checking whether bills were due and checking the financial statements. It seems a lot easier for such a person if the stuff comes in the mail.

If you've chosen to switch away from paper statements and bills, have you put any measures in place to deal with this issue? If so what have you done?