Why does the DMM make this circuit "work" ? (Power issues : servo & bluetooth)

i meant that the whole breakout module that the PCA9685 chip is on has just one pin (two in parallel actually) for the incoming V+, meaning that to insert a Pi circuit for each servo, one has to breakout the 3-pin male headers to achieve the setup of;

No.
You put the filter between the break out board and the servo.