A major reason that high performance racing or freestyle drones use LiPo batteries with high cell count (6s rather that 3s or 4s) is so that the voltage won't sag as much. This allows high performance drones to operate longer when pushing the limits of their power. Is there, however, any advantage to using a high cell count LiPo in a drone that doesn't pull enough power to cause any significant battery sag?
The advantage of running at higher voltage is that you can use thinner wires and lower current (i.e. cheaper) ESCs, as power lost to resistance is proportional to current (squared) and not voltage.
Another reason can be to reach higher prop RPM when you can't find motors with sufficiently high kV, or merely want to continue using the motors you already have.
Much of the discussion about increasing cell count is confused because people increase the total battery size, for example, going from 4s 1500mAh to 6s 1500. With 50% more battery it's not surprising that both flight time and the ability to provide bursts of power without voltage sag have increased. Similarly pushing more power through the same motor is not a free upgrade. Either it's using up a safety margin or the motors were larger and heavier than necessary in the original design.
Other than running high cell count batteries on low-power drones for the lols, there isn't much point in doing this, which will result in the throttle being SUPER twitchy and ultra-responsive to small stick movements without throttle correction.
Increasing the battery cell count on a multirotor results in an increase in motor RPM because kV rating of the motors means that RPM scales roughly linearly with battery voltage. This means that small throttle inputs will dramatically change the RPM of the motors and thus also the thrust output. Increased throttle twitchiness can be counteracted by a throttle curve in the RC transmitter or flight controller, which decreases the responsiveness of the throttle stick in-flight.
As you mention in the OP, the increased battery voltage will mean that less motor current is required to maintain stable flight (because electrical power is the product of voltage and current, i.e. P = IV), so the efficiency of the system will be improved because the decreased current requirements won't sag the battery voltage as much. However, larger cell count batteries are also heavier for a given Ah (Amp-hour) capacity. This means that more thrust is required to maintain stable flight to counteract the additional weight of the battery cells, eating into the efficiency benefits from decreased battery sag and possibly negating it altogether.