Here are what I see as the key points in this interaction (at least the IVR portion) that could have been improved:

This system required customers to login to a website before they could use the IVR.

Instead, segregate your customer service channels so that if one link in the chain fails, the whole system doesn't become useless. When possible, think of your IVR as a closed system. Some users aren't comfortable using web interfaces, and others may not have easy access to a local branch. If they're calling your IVR, they need service, and they think that this will be the easiest way. Let them leave feeling that it was easy.

This system put the caller on hold indefinitely.

Even if you must put a caller on hold while they wait in a queue, you should put a hard limit on how long the customer will be left waiting. If the timer runs out, then give the customer the opportunity to leave a call-back number where a customer service rep can reach them. Some customers may not mind waiting as long as others, and so you may wish to provide the option to "press 1 if you would like to continue waiting." But let the customer choose when to stop waiting, without feeling they've wasted their time.

This company only had one channel through which to provide user login information.

If callers to your IVR cannot provide the preferred validation information (e.g. info from the "packet" in Francine's nightmare), allow them to provide alternate information. For example, offer the option to be transferred to an agent who can go through various confirmation questions (e.g. mother's maiden name, address, etc.) to confirm identity. Depending on the security needs of your IVR application, it might even be possible to automate this.