It would have made business sense to settle if evidence (provable evidence) existed of the harassment. Just like it doesn't make business sense to settle if there is NO evidence. Otherwise, it just invites more lawsuits.
And once the NRA made one or more financial settlements after claims of harassment, Cain became a liability (and a potential future financial liability) to the company. At that point, it would only be prudent to end his employment, probably very quietly. My guess is that they just let him know that it would be best for all concerned if he "sought career opportunities elsewhere."
Au contraire. It DOES make business sense to settle for exactly the reason I stated.
They paid a maximum amount of 45K to settle. If they are right and choose to fight, 45K is about what it would cost them, likely more.
So, fight and spend 45K to win (or lose), or settle for 45K and no more time spent on it.
Good business decision to settle in that case.