LED sink drivers, daisy chained to a shift register...

Hi. I'm building a common anode LED cube. Seems to be a popular project, and great soldering practice for my son!

I found this excellent project, where daisy chained STP16CP05 (constant current sink drivers) are used to sink the cathode columns and the last one in the chain is used to switch some transistors that source the anode layers.
Schematic:http://www.hownottoengineer.com/projects/images/led_cube/LED_CUBE_sch.jpg
Full project:HNTE - LED Cube

He built a great project and was kind enough to make an excellent tutorial. However, I'm not sure how to use a constant current/ voltage-varying device like the STP16CP05 to switch a MOSFET. It seems like voltage (from STP16CP05 output) to the MOSFET gate would be varying while the MOSFET gate charges, since the STP16CP05 is a constant current device. His design certainly works, but I'm interested in learning rather than just copying a design.

So my question is, if I were to replace the last device in this chain (to switch the MOSFETs that source the anode layers), should I connect my last STP16CP05 to a latching shift register? For example, it looks like I can daisy chain a 74HC595 shift register to the end of the STP16CP05 chain. Output of the 74HC595 is only 35mW max, but I assume that would be enough to drive some MOSFETs like IRF7424 or IRF9328.

Does this sound like a good solution, or am I just complicating the design? I could just use another STP16CP05 as in the project at the link above. I already have enough STP16CP05 to do that, and I could use those to drive the MOSFETs if that is just as good. I would appreciate any advice.