Best way to implement score board logic using Arduino and Shift Registers?
There isn't really a best way (and such questions are usually discouraged as they can easily descend into argument).

Personally I would go for chaining all of the '595s together so you only ever use the 3 pins; it gives you the opportunity to use a smaller MCU like the ATtiny when you put it all on a board later which makes it (slightly) cheaper and (potentially) smaller.

From a software point of view, it doesn't make a huge amount of difference - doing it the way I have suggested means you send the 3 values out in the correct order every time one of them updates. Separating the 3 scores with their own dedicated pins means juggling which one needs to be updated each time. Neither option is particularly complicated and with this sort of application efficiency of code is mostly redundant because the entire MCU is dedicated to the (simple) task.