Hi and welcome,
You are correct, many '595 tutorials and instructables ignore the absolute max characteristics of the chip. Most of them get away with it because what probably happens is that if you source too much current, the register's output voltage drops, reducing the current and the led brightness. If the '595 is sinking current, however, there's a danger of overheating and shortening the life of the '595.
Your idea avoids those issues, but in effect you are multiplexing the leds, so reducing their brightness to one eighth of the brightness that a single led would be. Although each led receives 20mA while it is briefly lit, it is not lit at all for seven-eighths of the time, so its average current is 20/8 =2.5mA, so not very bright.
Your best bet for the brightest possible output of all 8 leds is to calculate your series resistors to stay within the 80mA (I seem to remember) limit.
Alternatively replace the 74hc595 with one of its higher-current equivalents, the tpic6-(a/b/c/d)-595. Note these can only sink current, not source, but much more of it. Or place a ULN2803 between the 74hc595 and the leds. Again, the '2803 can only sink current.
Paul