What you describe can work, and occasionally does. It's known simply as debt settlement, debt negotiation or re-negotiation. Whether or not you're able to come to such a resolution in your case often depends on who the creditor is, and what they're able and/or willing to agree to.
While your notion indeed turns a mutually bad situation (your credit score keeps goes down, creditor's not bringing in any revenue) into a mutually good situation (your credit score's repaired, the credit company gets paid) you're missing a couple of key issues from the analysis.
First, the longer an account goes delinquent, the borrower's credit score can continually get dinged, sure, but the creditor's outstanding balance continues to rise with the accumulation of interest and fees. Thus, the more prolonged the situation becomes, the more money the credit company is ultimately entitled to.
Then, creditors report to sometimes one and sometimes multiple different credit reporting agencies. Often, once the reports are produced, they're unwilling to go through the process of changing your past reporting. (The entire reason for having a credit score, of course, is to gauge a potential borrower's likelihood of remitting regular payment(s). Thus past reporting is, to them, just as important as current reporting.)
Thus, while it may be fair, and also extremely profitable to creditors, it's simply not how most creditors are set up to operate, and thus the procedure likely falls outside of the scope of established business practices for many creditors (think credit card companies.) While it's certainly possible, it simply doesn't happen all that often.
One of the primary issues you're going to run into is that the red tape at creditors is thick, as there are tons of different departments and it takes a significant amount of time negotiating to even get to the right department, let alone a representative with any authority. Then, typically that authority is only to process and then take a reduced amount to settle your debt, report it to the credit reporting agencies as "settled" and then issue you a 1099C for the difference as earned income. Changing credit reporting just isn't something that happens frequently.
You will receive some benefits to your credit score for having the debt listed as "settled in full," but it isn't the same as being "paid in full" and certainly isn't the same as having previous delinquent marks actually taken off of the credit score altogether.
I've never heard of a borrower having to go through an escrow office to handle such a transaction, though I suppose that that is a possibility. Typically an attorney can handle this for you.
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If you pay off an account that has been delinquent, any future credit reports will show a zero balance but will continue to show that the history of the account included being sent to collection. Also, the most commonly used credit scoring system, FICO, apparently does not differentiate between paid and unpaid accounts that were sent to collection. You are correct that this is irrational in that it dilutes the incentive to pay the past-due bill; but as the previous responder indicated, the collections system, especially on the part of the megabanks that issue credit cards, is just not set up for this kind of individualized negotiation.
On the other hand, the New York Times recently reported that another generator of credit scores, Vantage, has decided to ignore collections history for paid accounts -- not in the interest of being fair to consumers, but because history shows that late payment has little value as a predictor of future creditworthiness (which is, in theory, what the whole credit reporting and scoring system is supposed to be about). To the extent that this practice catches on, or even if a future prospective lender relies on Vantage and not just FICO, you will gain some benefit from paying this bill even without a quid pro quo.
I don't know what makes you think a creditor has a lien against your property. Normally this would only happen if they filed a lawsuit and got a judgment that you owed the money. Tax debts, student loans, and child support are exceptions to the rule.
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