Shift Register

Yes your boards are looking mighty fine there 8)

TheCoolest: Just as a proof of concept, I modified the design a little (sort of a patch, not a complete redesign) to include the header on the left side.

Reminds me - on the I2C versions, two connectors - as some designs have - enables easy chaining of displays (particularly but not exclusively when using the "Dupont" cables).

I2C allows you to daisy chain a display because there are only an SDA and SCL line. This module also has a CS line, it means that you will still have to run at least one line from the uC to each display separately. I haven't thought of this honestly, but the proof of concept board above will allow you do daisy chain power, SCK and MOSI lines.

With 74HC595, you need 3 lines - SCK, MOSI, and Latch (or SS) and you can daisy chain a bunch. I've done 20 of TCIP6B595 which is basically the same chip with open drain output. Limiting factor will be signal quality of SCK in the end, which goes to all chips in parallel.