Does the King Air actually land ? It doesn't appear to do so when it comes to calibrate London City.
Yes, small Citation business jets are regular, plus the very occasional exec twin prop owned by someone with more money for fees than sense to go nearby.
Quite some years ago (1990s ?) I also seem to recall that, one Christmas morning, when the airport loses most (though not all) of its scheduled flights, there was a whole group of light Cessnas etc landed for a few minutes.
Regarding looking at Concorde, this was a never-ending fact, it was even more prevalent at JFK, where on taxi out every ramp worker from every airline, and anyone near a window, was looking at you. And I can remember being on Cathay awaiting departure from Heathrow 27R one evening when Concorde came out from T4, passing us all without stopping, and the Cathay skipper giving us all a non-stop commentary on every aspect of it. Yes, it did shake up our 747 as they opened up for the take-off roll, and we needed to do another cabin secure call after it had gone from sight, to be sure everyone was sat down again ! Another recollection must have been about 1980, in the dining room of the old Excelsior Hotel on the A4, with departures from 27R somewhat visible from the windows. Someone shouted "look, there goes Concorde" and about half the diners threw down their cutlery and rushed for the windows. If you were in the Business car park on the northside when it went, that was the best, it used to set off several hundred car alarms !