Usage of 74HC595 for LED cubes: isn't it bad?

You are right, but many examples use either 68 Ohm or 220 Ohm current limiting resistors: at 5V that's way above the 8mA we should be constrained to in order to stay within specs.

Now, my question isn't "how can I use 74HC595 to drive my LED cube?" but "shouldn't we consider all those cubes faulty and bad design?" or, if you prefer, "how come those cubes seem so popular and don't burn out the shift registers?"