[thaumasiotes]

That link doesn't support your interpretation at all:

> If you try this approach on a credit card debt, it’s unlikely to work. Why? Because you probably won’t be able to show that the amount owed is uncertain. You’ll be held responsible for the amount that you charged—a number that’s likely relatively simple to figure out. On the other hand, if you’re working with a contractor and you disagree about how many hours it took to finish a job or the supplies necessary to complete a project, you might be able to satisfy this requirement.

The article makes no mention of a cover letter. They stress two things:

1. The amount owed must be subject to dispute. The hospital obviously satisfies that requirement.

2. The phrase "paid in full" was visible in an obvious manner.

That's it.

[bluefirebrand]

The cover letter illustrates that the amount is in dispute in writing. It is likely important in order to satisfy 1.

[thaumasiotes]

No, the phrase "paid in full" illustrates that the amount _is_ in dispute.

What you need is for the amount to be subject to dispute. See how the example they give of an invalid attempt to do this is a credit card bill, specifically because the amount of the claim is fully objective?

The cover letter doesn't add anything there.