Don’t worry, I went through the same thing trying to get mine to work. The 7219 (If I remember correctly) does drive common cathode displays, however it still wouldn’t be able to power the large ones. The problem is that the 7219 sources (can provide) 5 volts on the segment output pins, and sink (ground) 5 volts on the digit pins, because it operates at 5 volts, like the Arduino. The spec sheet for your display says it needs about 12 volts to drive the display, so your 7219 wont cut it. If you tried to put 12 volts into your 7219, your would probably fry it AND your Arduino.
The 7219 IS the easiest way to go, however there is no way to make it work with high voltage displays (and I tried!). The magic of the Allegro 6278 (and other drivers) is that you can communicate over 5v with the Arduino, but they’re able to handle much more voltage on their output pins. It acts like a bunch of little switches, except it’s a sink driver, as opposed to a source driver like the 7219. This means that you provide +12 volts on your common anode, Ground to your sink driver, and tell the sink driver which pins to short to ground, letting the current flow through whichever segment is connected to that pin.
The 6278 has a built in shift register - I believe the only real difference between using the shift register and using the 7219 is that the 7219 has the nice LEDControl library to take care of the dirty work, but really the shift registers are rather simple to use and should do the job nicely.