That was a strategic play on their part.
Their demand to "admit the debt" wasn't vanity, but a very wise legal strategy. If you admit the debt is valid, that restarts the clock on the statute of limitations. This greatly increases the cash value of the debt in the used-debt market. Huge win for them even if you don't promptly pay. Now they can sue you!
So this collector was brilliant. 90% chance you would have thought "what difference does it make since I'm about to pay it anyway?"
98% of the time when you try to engage with a collector via telephone, they will whip you. These are professional manipulators, trained to put you at the worst possible disadvantage in the law. Case in point.
Paying DOES NOT fix your credit!
I bet they implied that it would, but did not say it.
You suffer from a common misconception there. You think that a credit report is an "extortion" to force you to pay old debt. Not at all, and that would be kind of appalling, actually.
The credit report is an accurate record of how reliable you are for paying debt. You did not pay this for years, and that is a fact.
You almost changed that report to "did not pay for several years, and then finally paid". And since you would have acknowledged the debt (by paying it), that would have also restarted the clock on that credit mark. So it would be on your credit 7 years more.
Yes. Paying off old debt makes things worse.
Hey, I didn't design the system. I think it's a stupid system full of perverse incentives, that does not conform in any way to core human values of "be honest", "be fair" and "keep your word".
Since they have built this sick system and they game the system (in pursuit of filthy lucre above all), you too must game the system to get what is fair.
You must "make an offer they can't refuse".
It must be a written contract. The deal is "you pay, they remove the mark".
But, watch who your counterparties are. If you agree with the collection agency that they won't mar your credit, you don't have an agreement with the original creditor, who could continue to mar your credit. So you need to find out who actually holds your debt.
Anyway, the bones of an agreement are as follows:
Whereas there is a bill (identifying info enough to identify it, but no more) and saying HIPAA does not apply to this contract. That's throwing them a bone if it's medical related.
Whereas all of you recognize that there is a genuine disagreement as to the validity of the debt.
Whereas all of you desire a quick, final and no-fault resolution.
You all agree you pay $X to fully settle the matter. Paid in 30 days or entire agreement is void, but, statute of limitations restarts.*
Any payment whose purpose is unclear applies to this first.
This is the final resolution.
No-fault. Parties agree this settles matter in good faith. Settlement is for practical reasons, is not established as a valid debt, and any difference between amounts is not a canceled nor forgiven debt. **
Parites agree not to defame or disparage the other in any way, including and particularly that no adverse statements will be made to credit reporting agencies, and past statements disavowed.
Jurisdiction of law.
You can do this as DIY legal, but it might be worth the time of a contract lawyer to get it right.
* Never miss an opportunity to "throw them a bone" (or some raw meat in this case) if it doesn't put you at disadvantage. Since you will pay, it is moot.
** This prevents an IRS problem: When debt is forgiven (owed $1000, pay $400) the forgiven ($600) is considered taxable income for you, and they can send a 1099 to the IRS and the IRS will want tax. By this agreement, they can't. That would be tax fraud if the debt was genuinely valid.