LED and the resistors
Assuming that we have a LED with forward voltage and current 2 volts and 10mA respectively.

Yes that is true but this is only a theoretical thruth.

In the real world a 2 V, 10 mA LED does not exist, at least not for long. Measure it later and the current will be different. LEDs have an exponential relation between Voltage and current. Only a small voltage change will cause a huge current change. Adding to this, the behavior of a LED is very temperature dependent. A small change in temperature will also result in a huge current change. Also not all LEDs are physically equal. Even a number of LEDs made at the same time in the same machine will be somewhat different. One will draw 20 mA at 2 V but another one 15 mA, 115 mA, 130 mA and 1 mA. All very unpredictable.

So this means that when kept as 2 V the current through the LED is highly unpredictable. It can be much lower and the LED would produce less light but the current will often be much higher as well. When too much current flows the LED will be damaged.

The solution is not to let the LED control the current but to use a resistor. There will still be some dependency on temperature etc. but the resulting current variations will be kept within safe limits.

A resistor or some other form of current limiting is required. In some designs the resistor might not be present but might be there in a hidden form like a small battery which has a high build-in series resistance.