Is it possible to load values to a counter upon startup?
I'm having trouble seeing the pins on your parts, but I believe I see a 'clear' or 'reset' input.  If you place an RC circuit that goes into a Schmitt trigger and run the Schmitt trigger into your clear outputs, then at power up, the resets will all be held until the RC charges, typically a few milliseconds.

Looks like fun!

Edited to add a schematic using a comparator instead of a dedicated Schmitt trigger.  Same idea.  Set the delay using R1 and C1.  Set the threshold using R2 and R3.  If you need to invert, then keep R4 where it is and reverse the voltage divider and the RC circuit.







simulate this circuit &ndash; Schematic created using CircuitLab