Your PSU is not strong enough, this causes the issue.
During Power-On, the PIC requires most of the power. For these 1..2 seconds the voltage probably sinks below 4.6V, and unexpected things will happen - or nothing will happen…
I would propose to try it without LCD backlight first, and maybe only with a single PIC (remove the slave PICs from the sockets) just to check if my assumption is correct.
It helps you to determine, which part or module consumes most of the current. E.g., for a single core module w/o LCD backlight and without DIN/DOUT module I would expect a current drain of about 50 mA, with backlight maybe 150..200 mA - this can be measured at port J2 (if you are using the optimized PSU circuit) or at J1 (if you supply the core module on the common way). You need to disconnect one cable (use the +5V line), and insert the multimeter there as a “bridge”.
By connecting the DIN module, the current drain shouldn’t increase - if it increases, it indicates that there is a short somewhere around the 74HC165. This can be visually checked at the bottom of the PCB (search for soldering clumps, etc.)
I had the problem with the DINs once. I had plugged the cable from the core to the DIN in the wrong way around… I’m pretty sure you checked that already though…