The "shift registers" is nothing more than "parallel to serial" conversion. If at one end, I had the BCD version of each digit loaded into a shift register. No big deal, right. So I have 5 nibbles of 4 bits. Call it hex, call it octal. I don't care. It's 16 bits. Fixed decimal place.
Octal fits in standard UARTS. Maybe they packed it into 2 UARTS and they clocked the data out.
Maybe they had their own 16 bits of info, created a start bit and a stop bit, and sent the data out asynchonously.
The whole point is to use a few wires as possible to communicate to the display. Using existing wires or wireless is even better.
What kinds of signals do you need at the display end?
Where is the engineering problem with the current system?
How much does the current system cost?
What are they willing to pay for a retrofit controller?
What are the typical wires that are used for communication? Both type and function?
The simple circuit could be part of it too, but with the prices of microcontroller so low and glue logic so hard to find why not go microcontroller based.
Although I might not like it, because of expense, it's surely possible to design a sign system using say a Smart relay type of PLC and a MODBUS slave device.
https://www.electro-tech-online.com/custompdfs/2011/06/14105EN.pdf and a little bit of glue.
It won't be anywhere near cheap, but it could be easy to do.
This
https://cubloc.com/product/08_01.php isn't too bad. It had MODBUS and programmable in C, so it's a possibility.
Maybe add a 16 bit MODBUS output module like this one:
https://www.omniflex.com/otspecialfunctions.asp You have 16 digits x 4 displays. You may not be able to multiplex that fast, so your glue might be a module with a latch for each digit and a brightness control and whatever power board issue you have.
I'm just thinking outloud here with not much to go on. Also check out
www.cpustick.com for some interesting wireless options.