Why do LEDs require a constant current source and not a constant voltage source as power supplies?
Oh, you can drive an LED with a contant voltage. It's only that this voltage has to be in a very narrow band, depends on chip temperature and is not easy to regulate.

Look at the I/U chart of a small power white LED. See how the current varies between 1µA and 10mA (nominal current) in the small band between 2.25V and 2.5V? And how much it depends on temperature? And how the power and the excess heat also vary 1:10.000 ?

So, it's much simpler to look at the current directly. That's why the current is measured and regulated (or simply limited with a resistor.)