> So in theory, if you can pay $1,000 to buy your own debt, somebody else should be willing to bid at least $1,001.

Not at all true. Suppose I have, for instance, a debt dischargeable in bankruptcy, which has a current balance of $10,000, and $1,000 in assets. I might be able to pay $1,000 to purchase and settle the debt, but I'd have very little reason to pay $1,000 on the debt without it being a full settlement, and there's pretty much no reason for any third party to be willing to pay $1,001 (or any value not substantially less than $1,000) to purchase the debt.