My latest video can now be found on YouTube:
I have validated that the issues I was having with the zero detect circuit was caused by the logic levels being pulled down by the LEDS on the output of the ALU. I went ahead and replaced the LEDs with a small circuit board populated by an ULN2803 darlington transistor array to drive the LEDs. This solved that problem.
Otherwise, I have built the program counter and it works with no issues. The circuit for it is below.