Using serial-input latches from Pi to drive 24V load (LEDs)
simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. OP's first schematic rearranged for clarity.


  I wonder if there's a grounding issue here, I keep seeing mention of pull-down resistors but always in the context of inputs.


There's a grounding issue alright, but it's not to do with pull-down resistors. You have no ground connection between the two circuits. Connect the 24 V negative to the ground of the rest of the circuit.