Explanation of the circuit in an LED bulb
LED bulbs usually use a current source. That is a driver that provide a constant current through the LEDs. Putting the LEDs in series is then the logical way to connect multiple LEDs to the same driver. 

The reason the leds are put in in a series string of parallel pairs is to add some redundancy if one LED fails and to avoid mismatched LEDs making one string brighter than the other. 

If you short out a single LED the voltage across them all will be dropped by the driver so the current matches what it was programmed to emit. The matching led will not light up anymore.

The reason it started blinking when you replace an LED with a resistor is because the driver will have open circuit detection and reset itself if it cannot supply the voltage needed for the current it wants to drive.