A cash offer will still fall apart if the appraisal comes in too low, I would think. Or does the term "cash offer" imply that no appraisal will be done?
There is no purpose for a cash buyer to pay an appraiser to appraise a house. They have already decided to pay the price, so who cares what an appraiser thinks?
Probably the latter, the appraisal is typically mandated by the bank issuing the mortgage.
Appraisals are generally done to ensure that the loan guarantor will be able to recover their investment if the party taking the loan defaults. Therefore, cash offers don't perform appraisals. Cash offers lower risk for the seller and provide fastest time to close the transaction.