Is the skin effect a factor for low-current AC applications?
Is the center of the conductor beyond that depth effectively
  pointless, regardless of current?


No, the centre isn't pointless because, as mentioned in a RedGrittyBrick comment, 63% of the current will flow in the skin (if the cable is big enough) and still leave 37% flowing closer to the conductor centre. For resistance critical applications this is significant in my book.

However, if the skin depth were only 0.1mm (as per 400kHz) the central region (say +/-0.25mm) of a 1mm diameter wire is a little pointless. 

If you needed to pass large currents (tens of amps) you would make the wire bigger in diameter (enclosing more cross section at 2 or 3 skin depths) or make the wire hollow to save money.

It doesn't have to be massive current though. If you have a highly resonant tuned circuit, you want to keep series resistance to a minimum so you'd go for a wire that is unconventionally large to keep the Q of the circuit high.

I've seen a metal detector in a timber saw mill use standard copper pipe for its oscillator coil even though it was probably only passing a couple of amps at 100kHz oscillator frequency. They use MDs on timber in Canada to protect the saws from getting broken or blunt when cutting through timber that someone has been using for target practice!

For similar reasons I've used 750 strand litz wire to pass power from a resonant magnetic field to a small circuit located up to 40mm away. With 40mm distance the Q needed to be really high to create sufficient "depth" in the mag field at that distance. 

The strands were spaced apart in three coils of 250 wires each because not only was I trying to avoid skin effect, but also proximity effect, another problem with high frequencies not utilizing the full conductor cross-section.