For the schools in which the SSAT scores have been submitted, there will be no benefit overthinking the decision.
For the schools that SSAT scores that have not been submitted yet, I feel that for those who are applying “full pay” and live in good neighborhood, there maybe certain level of expectations from schools that the families would have the means to navigate through the on-line testing format of SSAT (or ISEE). If the scores are not submitted, the admission officers could possibly deduce that the scores are not that great unless there is other reasons given.