Neon lamps are gas discharge tubes and are not very capacitive at all. Neither is a switch. Typically a neon and large-value series resistor are wired across the switch, so when the switch is off the neon illuminates due to the lamp bulb filament conducting that small current without getting hot enough to light up. Swap the tungsten bulb for an LED bulb and the LED will light up with the small neon current. Once the neon is old it will flicker and cause the current to fluctuate, thus causing the LED to flicker too.
The flickering can also be due to the SMPS inside the LED lamp, if it has one. The smoothing capacitor of the SMPS gets charged until the voltage is high enough for the supply to start up, which will discharge the capacitor.
I am actually going to use this unusual high sensitive of the led bulb and increase the capacitive current by shunting the switch with a y-type cap. A bit more ligth from it is actually very helpful when i have to climb up the stairs to my house in a pitchblack night. No stumbling in the dark anymore before i can switch on the light.