OP your circuit is very close to LED nightlights. Actually difficult to design.
The input needs a say 100-220R flameproof, fusible resistor that tends to fail open if you power up at a mains peak, the inrush is quite high.
The
dropper capacitor is generally huge for ones rated for the application.
It's better to use the zener at say 9V or 12V and then a resistor to drop that down for the LED. You can use a much smaller cap and get no flicker, compared to being right across the LED. The electrolytic cap's ripple current is high and can shorten life.
Without any clamp zener, I had an LED fail open-circuit and the capacitor exploded
What I do to replace neons, if flicker is not a problem, is a resistor and LED with reverse-diode across it. NOT SERIES diode, that fails fast.
For no flicker, a bridge rectifier is good enough for 100/120Hz harder to notice, and just a resistor for the LED. Even 1mA can be quite bright with high efficicency LEDs.