I could see a third option as a "lemon detector". That is, instead of their output being a number, it would be a binary, intended only to limit downside risk in the worst cases.
I think in practice this is indeed the role they play. If I'm "overpaying" by 20%, the appraiser will probably still "write down the predetermined number", but if I'm overpaying by 100% and taking out a mortgage to do it, there's a pretty good chance that the bank's appraiser is going to sink that deal.
That's basically what they do, they're not to find the actual value of the property (that's more properly a tax assessor, perhaps) - they are there to make sure the valuation is "in the ballpark" to what it would liquidate for.
An appraiser will never get in trouble if a bank forecloses on a $500k property and it sells for $450k or so, but they will get in trouble if the $500k property only sells for $200k - they'd need to explain why.
I think in practice this is indeed the role they play. If I'm "overpaying" by 20%, the appraiser will probably still "write down the predetermined number", but if I'm overpaying by 100% and taking out a mortgage to do it, there's a pretty good chance that the bank's appraiser is going to sink that deal.