The closest handbook I have to the '31 NEC and it's language is a 1940 by Arthur Abbott. It does not reveal any reason behind the 6 throws.
A 1929 Handbook by Terrell Croft merely reflects the 1928 NEC requirement of a disconnect for each set of service conductors. It does, however, offer a clue:
The Service Switch Must Be Externally Operable Only When The Current Is Separately Metered And Whn Such Separately Metered Circuit Is Not Provided With An Externally Operable Switch And Enclosed Cutout- If the entrance circuit is provided with only one meter, the service switch need not be externally operable. Or, if the circuit is provided with more than one meter and each separately metered circuit is controlled by an externally operable switch, the service switch does not have to be externally operable. The only place where the "Code" requires an externally operable service switch is where a circuit is separately metered and where each such separately metered circuit is not controlled by an externally operable switch.
Meaning: If a meter stack does not have externally operable disconnects (i.e, you have to open a box to turn the power off) then the main must be externally operable (i.e, a handle on the side of the box). If each individual circuit (feeder in today's lingo) is externally operable, then the main does not have to be externally operable. In the language above, a 40-unit apartment building would either require an external handle on the main, or external handles for all 40 feeders.
So my guess is there was much confusion about 'externally operable', and what it would take to completely shut down the power in a building in an emergency. To simplify the process, a maximum of six disconnects was allowed. Why six? Probably just an arbitrary number.