I attended a Test Innovator seminar in October in which there was a panel of independent school admission officers. The panel generally comments that a lot of parents have been thinking about this standardized tests for 1 - 2 years. Parents are motivated to push the kids for good scores and then send them. They do believe that the weight on the standardized test will be less this year, but the tests could still be a determining factor as they help to validate the other parts of the application.
In conclusion, the panel feels that it depends on the circumstances such as how many applicants, are there students with similar profiles but one with score while the other does not etc. If the SSAT is submitted, it will be used as part of the holistic assessment.