Can I restrict packages to start on a fixed day of the week?

Example, when you only take bookings from Saturday to Saturday only.

Sorry, no. 

This fixed-day technique is sometimes used in domestic leisure markets, during peak tourist seasons, but not something you find in most business traveler booking engines.

Digital nomads are much more like business travelers than leisure tourists.  Digital nomads need much greater flexibility in dates usually.

Saturdays and Sundays for instance can be normal working days in some cultures.  And with time zone differences around the world, you can find Friday nights, Saturdays, and Sunday nights in one country can be normal working hours in their employers/clients offices somewhere else in the world.