Why could Rosyth support ships that Scapa Flow could not?

Did most of the Grand Fleet move to Rosyth after Beatty took command?

No. Scapa still held much of the fleet, while Rosyth's advantage was its greater repair and maintenance infrastructure.

The Grand Fleet did not simply abandon Scapa for Rosyth after Beatty took command, because Rosyth could not contain the entire force. As wartime wear increased, however, ships needed more repair, refit, and support than Scapa could provide. The same division of functions appears in British planning for Singapore, which was intended as a logistics and repair base on established shipping routes rather than merely an anchorage beside the expected combat area. The comparison distinguishes a forward operating position from the theater base that sustains it through a long war.