Yes, you could in theory, but you would be getting into borderline RF territory.
At "low" frequencies, I don't think a single ended (one wire) path would get you anywhere close to the kind of delivered power (5 mA through an LED) you are asking for.
As far as "run one wire to a chip with some rectification/filtering circuitry and LED, and power it" -- no, not unless you create a secondary two wire power supply for the chip and LED system tapped off the one wire power supply.
Why did you want to do this? I think the reason you don't see it in practice is because there are simpler and more practical engineering solutions to the problem of an isolated power supply.
To visualize how such a circuit might work, consider the antenna section of a radio receiver. The antenna couples an AC signal into a coil, and an output signal is tapped off that coil, filtered, detected and amplified. However the antenna signal is tiny and delivers very little power.
If you took the same basic circuit and coupled the AC signal into the "antenna" through a capacitor rather than by radio waves you could get the result you are looking for. But I think it would be cumbersome and inefficient.