Why don't I see any examples or tutorials using a bunch of PNP transistors (differences to MOSFETS?) between the shift register and the LEDs to handle the current?
Two reasons:-
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Depending on how you scan the matrix you might have a ULN2803A sinking 8 LEDs but only one pin sourcing the current for one LED at at time. As described here:-
Arduino Workshop -
There are a lot of moron tutorials out there. Some don't even use resistors to limit the current in the mistaken belief that because it is multiplexing there is no need to. This is wrong. See:- LEDs