I don’t understand why you would PIP a person and keep them waffling around for months on end collecting paychecks when you can just severance them out and pay less in time and money?

You are probably right... in most cases. Companies have lawyers and those lawyers are very risk adverse. The PIP is basically a period of employment where the company collects evidence on your (lack off) performance. This evidence can be shown as justification for firing, in the unlikely even it is needed.

PIP = "Paid Interview Period"

I've never seen anyone recover from a PIP.

We do it because we legitimately want them to improve, and the last resort is "if we can't work it out by X, we can no longer employ you". I don't think we've done it more than once or twice, though, because increasing the directness of the feedback that the employee is not doing well has generally worked.

It is to cover some of their bases in a potential wrongful termination case

Management has its own incentives to limit turnover and maintain their department headcounts.

I believe that often times the PIP is an attempt to get more value out of an employee that you would otherwise replace with offshore resources.

i.e. "Maybe if we spook them, we can get the work of two people out of them for a while."