Rate law of the iodine clock reaction
IMO, the only reaction that you are studying is the first reaction:  the conversion of $I^-$ to $I_2$ by hydrogen peroxide.  The thiosulphate reaction is simply a dramatic way to indicate when a particular amount of iodine has been produced.

The reaction was not slower when you doubled the amount of thiosulphate.  Twice as much iodine needed to be produced (and reconverted to iodide ion) before the thiosulphate was consumed and the first permanent iodine changed the starch indicator blue.