Controlling Power LEDs through multiplexer
The datasheet for the CD4051 shows that both ON and OFF there will be +/- leakage current on the output pins. This tells you that the voltage present on an output pin can be EITHER high or low depending on the leakage current. It is no '0' which is a digital state.

Your addition of the 100 kohm resistor simply provided a path for enough of the leakage current to ensure the output was low enough to cutoff the FET.



You would be much better to use a digital device such as the 74HCT595 serial shift register to achieve your goal. With this device you can serially shift out the state required for each of your LEDs and then transfer from storage to output using the STCP transition. You'd need only 3 MCU pins to do all this and with the /OE set low you get definite high and low signals for all channels.

One potential negative is that the 74HCT595 does not work at 9 V ….so this would need to be powered from the Arduino 5 V supply.