Thanks for your reply. I did look at the data sheets for the MAX6654/6655 type chips, they seem quite a bit more complicated to use than the 7219 because they try to be so much more versatile; more LED arrangements, a built in port extender, etc. I also don't think I can get them in through-hole which might be a problem for me.
Those chips also support 128 LED segments which is one reason for the complexity. From your post, can you point me at any drivers that do 1 or 2 digits? I only need 4 digits so that won't be so bad for me.
Using 2 shift registers, a transistor array, and resistors will up my component count a lot. Programatically, I'd use a timer interrupt handler to do the multiplexing so I could have a set-and-forget interface which I find a lot cleaner than blink-without-delay, but that whole scheme seems so inelegant. I'm just surprised it's such an easy thing to do with 7 segments but so hard with 14.
I thought the 14-segment display would be a pretty solution, but it's at the point where I'll probably just go with a character LCD.