When the sacrificer is specified in a different way (with formulas like "the one whose son was just born should perform the ritual …"), the desire is no longer needed by the Veda. It is, on the other hand, still needed by the person to engage into the action.
How do these two points harmonise? If desire is needed, why do people engage in sacrifices, who seemingly do not bear any result? Rāmānujācārya (Tantrarahasya IV, §10.5) can explain away most cases since he says that in the viśvajit and in similar sacrifices, where no specific person is prompted, one has to postulate as sacrificer someone who is desirous of haven. In fact, since 'heaven' means happiness according to the Śabarabhāṣya, this specification will suit everyone. In this way, one has satisfied at the same time the need for a specific sacrificer and that for a result to strive for.
But what about the case where a specific sacrificer is supplied, but no hint of a desire is left (see the formula mentioned above)? In order to make sense of these cases, Rāmānujācārya (and other authors?) has to abandon his usual Mīmāṃsā 'down-to-earth' attitude. In §10.11 he answers that the very non-performance of a prescribed duty is, for cultivated people, something one desires to avoid. "In fact, dharma is also one of the human purposes (puruṣārtha)". In saying so, Rāmānujācārya interprets artha as goal and connects it automatically with one's natural desire.