How can I see the number of test scripts that have not been run in the UI on test cases?

A test case is typically associated with one test script unless the execution is required to run in different test environments for the same test case.  Please refer to the test case and test suite overview in the knowledge center especially on the Test cases with multiple test scripts section.

Anyway, assuming that there is an execution record already created for each test script, you can check from the Test Case Execution Records list view.  Make sure that the "Test Case", "Test Script" and "Last Result" columns are displayed.  Set the column filter to "Not Run" for the "Last Result" column, run the query then change the table view to group by "Test Case".  You will get the test case execution records that are not run with their associated test scripts of the test cases.