Full auto would require charging a huge version of a similar mechanism for a single volley and as a non-actual-engineer, I do not know that it is possible to output the torsion energy in a controlled manner preventing the gun from exploding violently.

edit: But, yes. This is more akin to a revolver than to a machine gun(or even chain gun as Wikipedia implies).

Pulley and a big rock to precharge. We know they had the components since catapult existed.