Does running LED at higher voltage damages the LED?
The idea sounds good but won't work. LEDs cannot magically adjust their incoming current to suit the prevailing applied voltage unless it has some type of switching regulator built into it.

On the understanding that it doesn't, if you increase the voltage a little bit across a partially conducting LED, it will draw a lot more current and quite possibly enough to near-instantly fry it.

Take the example of a 24 ohm resistor. With 12 V applied it takes 0.5 A but with 19 V applied it takes 0.792 A. There is no power regulation implied in ohms law.