There very well may be a better mechanism, but an overnight Scheduled Workflow Rule for User may suit your needs.
The function GetWorkingTimeForGivenPeriod() returns working hours for a specified period. Consider the below logic, if today's hours are zero and not a weekend....
(Note: you could add a placeholder user's calendar to check for other calendars which may indicate a non-working day. The more complex your organization use of calendars is, your logic here might need to be more sophisticated.)
Then...Create a new Timesheet:
And Timesheet is created:
Also, from the Timesheet View, see that the column of the 11th is grayed-out because its a non-working day for this user per it's individual calendar. (I'm pretty sure this is standard Clarizen functionality.)