Credit scoring models vary, the latest FICO 9 and VantageScore 3 ignore the $0 collection balance, so it could help your score to pay it off. Under older versions (still commonly used) it might not help your score to pay them off.
For example, under FICO 8 and older there is no benefit to paying off an account in collections (from a credit score perspective):
As far as your FICO score is
concerned, two things are considered; has a collections appeared on
your credit report, and when it was reported. So whether or not you
pay your collections off is really a personal decision.
The collection account will also typically have less impact over time, but this varies by scoring model.
Can these companies affect my credit score post my debt being sold to them?
Consider the charge-off/collections to be the harm. They don't cause additional harm beyond that, but continued harm. They don't get to start reporting new late payments, but the collection account will continue to be reported.
Why would I pay these collection companies if it would not help out my dying / dead credit score?
You owe a debt, non-payment can result in legal action to compel you to pay. Depending on your ability to pay this might push you toward bankruptcy.
Additionally, many lenders don't just use credit score, they use all the information on a credit report (and more). A paid off collection account looks better to a potential lender than one left with a balance whether or not it affects the credit score.
Experian - Collections on Your Credit Report
Credit.com - How to Remove Collection Accounts From Your Credit Reports
Businesscreditreports.com - Facts about FICO 9