do mortgage companies ever negotiate a payoff if the payoff is cash

Generally no. The reason? A mortgage is a loan to you but an investment (and income stream) for an investor. When people pay off their mortgages, the interest rate might be a driving force, but not cash. Most people pay off their mortgages, (refinance), when rates are low, so the investor (or bank), can only make a new loan at a lower interest rate, which is less income. This was, generally, the reason to create mortgage back securities, but that’s a whole other topic.