The idea behind starting is to pressurize the combustor section in order to hold the fire in the burner and to expand through and accelerate the turbines. There is not enough horsepower available either on a starter or in windmill to provide this pressure rise by spinning up anything other than the relatively small high pressure compressor. The fact that the other spools do turn is associated with the exit flow generated by the high pressure spool passing through their turbines.
In fact the burner can light off and hold the flame inside in some cases without the other turbines turning (locked up mechanically). However any attempt to accelerate may create a mismatch in pressures resulting in an aerodynamic stall in the compressor.
Starting should have less to do with how many spools than the burner characteristics.