For Billable time you should be able select a specific employee to attach that work to.
You can. Read the portion of the Guide below having to do with adding custom fields:
You can do this as in the example as a free-text single line field where anyone can write in any name. Or you could create the custom field as as dropdown box and define employees. You can then sort the Billable Time ledger by employee or search by employee if that would be useful for creating payslips.
The fact is, most things in Manager can be customized for your needs with custom fields. But the underlying philosophy is not to bloat the software by default with things that are not essential for all.