Hmm. I suddenly got worried about my aging memory banks, so I just pulled down the SN74HC595 datasheet from TI. It says +/-6mA output drive at 5V on the front page! I think that's where I got my 5-6mA figure -- it stuck in my mind that way.
It also gives an absolute package maximum of 70mA continuous and a continuous output maximum of 35mA. But, as they also say, operating at these absolute ratings is wrong-minded -- they are only to tell what maximum stress they can tolerate before being damaged, not operate.
Did I miss something, then?
Jon