This is actually a fairly standard way to design a clinical trial, called the "stepped wedge" design. Everybody gets the control arm, followed by the treatment

This is actually a fairly standard way to design a clinical trial, called the "stepped wedge" design. Everybody gets the control arm, followed by the treatment, but the time of the switchover is randomized, often in steps (hence the name). If people consistently recover at the same time relative to the treatment, but at different absolute times, then you can conclude the treatment works well.