I was wondering what was wrong (if anything) with wiring it up as in my illustration.
There's nothing wrong with the series aspect of it, as long as either A) all the LEDs are fairly well matched (i.e. have about the same luminous efficiency) or B) you don't care if some of the LEDs are brighter than others.
The parallel part is dicey, though. LEDs are current-operated devices, and one LED/string of LEDs can end up hogging most of the current, resulting in the others being considerably less bright or not visible at all. Problems with LEDs in parallel are much more likely than with them in series.
It hurts nothing to try, though. If you run into problems, you can likely juggle the LEDs around until you get them fairly well balanced. (There might still be differing characteristics across the temperature range, but you won't know until you try!)
Thing is, high quality resistors aren't needed for LED current limiters, and when you can get regular ones for literally pennies each, avoiding them doesn't buy much versus the potential problems. Unless one is trying to fit a project into a tiny space or designing a consumer product where even a few extra cents can screw up the profit margin, it's usually wise to give each LED (or series string of LEDs) its own resistor.
Which is why, as you noted, you see individual resistors so often.