1.3 is more than arbitrary, it insures that sufficient energy (speed) is retained during normal speed deviations in normal conditions. In turbulent conditions you can go rapidly from stabilized approach to stick shaker or full stall instantly, in which case an approach using 1.5 may be called for.
Conversely, in smooth air you can fly a 1.1 approach to minimize the runway used.
Rather than focus on speeds you would need to look at this first from the perspective of AOA (Angle of Attack). In quite a few aircraft speeds aren't even used and AOA is the primary speed indicator for approach, landing and other phases of flight. This allows the aircraft to be flown with sufficient stall margin at any weight in any condition (Flaps up, down, engine out...pick one), however this does not provide you with any way of knowing (without calculating weight and speed) how much runway will be used for the landing.
In larger aircraft the stick shaker/stall sensor are in fact getting inputs from an AOA probe. AOA is the most accurate way to instantly check where you are flying in respect to the wings performance.
The use of speed allows operators to calculate (based on projected weight) predicted performance in respect to the amount of runway required for landing. Weight will be based on performance generated during flight testing from many factors including AOA. This for legal reasons is perhaps the easiest method to insure suitable runways with sufficient runway length are selected for the conditions.