I wonder if gliders that are powered by electric motors only during takeoff and landing could be a solution here. It's not like we get very annoyed by eagles soaring in the sky.
Takeoff and landing are probably always going to be the noisiest part thrust-powered flight.
That is a good point... So now I'm back to the giant slingshot idea.
What about a buried rail gun, as a less explosive version of the columbiad in Jules Verne's From the Earth to the Moon? You'd step on to a circular platform, get slowly lowered into the barrel, then shot into the sky, where the wings would open :)