Suppose you got an unsolicited email spam with the following offer. You deposit money in the bank account of some unknown website owner and he'll use that money to pay your bills. Naturally, there's an attractive convenience. But I suspect very few individuals would accept the offer.
Now suppose the same entrepeneur approaches your bank. As part of their deal, your bank agrees to brand the service. When you log into online banking, there's an offer to "use our online bill paying system". The link goes to a website that displays your bank's logo, and everytime you fill out a short form, money is withdrawn from your account.
I discovered with my local Credit Union that other than the superficial appearances, there is no difference between these two scenarios. $450 is missing from my account. My Credit Union says I approved the transaction and any problems are my responsibility. As part of the discussion, the CU now acknowledges a 3rd party bill paying service that is completely independent. But otherwise, none of my persistent questions were answered.