Two things show up here:
1) all script steps that start with "insert" fail to work unless the field specified is also present on the layout that is current at the time the step is executed. You get a "silent failure" if it is not. (Silent Failure: No error message is displayed, you only know that an error occurred because the field isn't set and an error code is returned.)
2) You are creating a new record after inserting the current time. Thus the record you are looking at when you check results is very likely not the record modified by your Insert Step.
Use Set field instead of Insert:
Set Field [ Log Start::StartTime ; Get ( CurrentTime ) ]
Note While this step does not require that the field be present on the current layout, it STILL requires that the current layout context be such as to make modifying this field something that is possible.
Implement this one step at a time so that you can see how each step affects your results. Start with just the set field step while on a layout based on the Log Start table ("log End" is based on this table as shown by the name in parenthesis.) Then add the new record and Go To layout steps one at a time so that you can see how they affect your results.