This is marked as a desktop topic.
This is so Easy, in Desktop. Yes, you start from the Timesheet. Set the Employee Default, and each existing Employee name, to be Paid from Time Data.
Now you see two columns: Payroll item and Service Item. This will tag the Service Item with the payroll cost, when you Pay from Time Data.
Your misunderstanding of Timesheets is something I just explained in detail here:
You need to set Preferences to Job Track per Earnings item, and Job Track Expenses, as well.
All of this is so easy. It starts with Timesheets.
I also have an Image attached for that setting and will make others with examples of the many reports that show this works Great.