Welcome to FlyerTalk, Tushar.
I'm truly sorry you had this experience. You can request compensation, but IMO they practiced proper risk management andvfollowed the letter of your pediatrician's original note:
we had been provided with a note by my son’s pediatrician indicating that the airline allows us to carry his nebulizer medicine on board for his asthma treatment. The medication was to be given only if necessary.
my wife inquired with the airline staff if we can use my son's nebulizer medicine on board if needed and if the flight had a an electric outlet for us to administer the 10 minute treatment, only if needed.
"only if necessary" and "only if needed" are huge words to digest in this context; they envisioned necessary but unavailable treatment, causing an emergency descent to an intermediate airport, medical intervention, delays to all passengers and possibly arrangements to service the airline with fuel filing new flight plans, possibly infringing on available crew flight time and disruption of thevreturn and possibly othervflightscDispatch had programmed for this aircraft.
There's no way the airline's flight surgeon / consulting physician to determine more without a full medical evaluation, probably securing the pediatrician's records or telephone consultation; they don't want to do that for obvious reasons, and if they had the results might have been the same: flight gone.
I'm sorry this all occurred, but in trying to cover all options you set off a big warning bell. And you might not actually have wanted to risk that the ultra-dry cabin air in an aircraft pressurized to ~8,000 MSL might have triggered an asthma episode (and no ability to use the nebulizer because of the lack of power).
Cabin power, even if provided, is not sufficiently reliable to count on anyway. People with POC (portable oxygen concentrators) must carry sufficient batteries to power the device for the duration of the flight - and IMO some reserve time.
Good luck with your vacation and return trip.