How much of a discount that's worth depends on the likelyhood of collecting after the legal process as well as how much of a reputational hit going through the process would be.
I wouldn't negotiate with most things after the service was performed (unless it wasn't to my satisfaction), but most services will have a firm quote before the service (sometimes that's legally required). A mechanic can keep your car until you've paid, but a hospital can't refuse to discharge you until you've paid, so they don't have the leverage to get immediate payment.
(probability of recovery) * (1 - profit margin) * (amount owed)
today, for the right to sue on you behalf?
Note: I think the current debt collection system in the US is terrible, especially for medical debt. I suggest looking at the charity "RIP Medical Debt" if you do too.
At the very least, when selling a debt to a collections agency, the (re)seller of the debt should be obligated to first offer the same price to the person that owes money.
That seems reasonable at first glance, but it would lead to an explosion of people simply refusing to pay and waiting for the 'debt collector price' offer to arrive.
Depending on the debt, getting most of the difference between the legal expenses and the debt itself is in reach, given a creditor who is a rational actor.
Suing people who don't pay their bills is a losing proposition. You will be paying lawyers and getting very little to nothing in return.
In the "of course that exists" category, there are services that give a score about somebody's ability to pay.
I really hate how every business process can be (and frequently is) optimized to the nth degree at the cost of privacy.