Would you recommend hospitalization for surgery for a large papillary fibroelastoma or atrial myxoma discovered on an outpatient echocardiogram?

NOT necessarily to hospitalize if incidental finding given that both of these lesions are typically present for years prior to diagnosis (unless of course you are concerned about getting sued if a bad outcome occurs). If these lesions are found in the context of a recent neurological syndrome then certainly expedited workup is in order and sometimes hospitalization is the only way to accomplish that.