"My pay week starts on Friday and the timesheet will not let me change it."

That isn't the point of the Timesheet. Notice the wording in QB: Select Friday as the First Day of Work Week.

Work Week = the State regulation, typically used for determining Overtime, such as 7 consecutive Days.

The Pay Period is not the same at all. The Pay Period can even be, for instance 1st-15th and 15th-end of month. The Payroll will pull the data from the specific timesheet by date, not by the View you use for Weekly View of entering that data.