No you could in fact use a LED as an element in your rectivfier and then place the current limiting resistor across it.
The issue with that is the diode would only be lit up half the cycle and the energy wasted on the other half of the cycle. The result would be the LED would be half bright and half energy wasted as heat.
That said it would work and the led would be smooth as the frequency is likely high enough not to be evident directly.