The skin effect is the effective reduction in the cross sectional area of a conductor that carries high frequency AC. The current density is higher on the outside of the conductor than it is an the inside. The actual resistivity of the conductor is still uniform throughout.
Its essentially an example of back emf which is strongest in the middle of the wire produced by the changing magnetic field of the AC.
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.
You should use the finest magnet wire with many multiple strands or LITZ WIRE to improve L/R time constant and peak current for lowest loss cable. Your discharge rise time will then drop to xx picoseconds. But for conductor pairs use twisted pair Litz wire to reduce CM emissions.
Otherwise, you can jam many radios/mobiles with high rep rates of > 1pps affecting AGC. !!
In most conductors R(f) rises fast and L(f) drops slow from skin effect. This R effect increases more with iron content since it is from Eddy Currents. In DSL and cable modem skin effects change Zo, phase shift and group delay.... Tony
Although the geometry is different, a twisted pair used in telephone lines is similarly affected: at higher frequencies the inductance decreases by more than 20% as can be seen in the following table.
Characteristics of telephone cable as a function of frequency
Representative parameter data for 24 gauge PIC telephone cable at 21 °C (70 °F).
The spectrum of your pulse is not at all like a square wave, since it is not repetitive over a small interval. It is a continuous spectrum rolling off similar to The null of 2nd harmonics of the equivalent pulse of a "square" wave and then rolling off above the 0.35Tr rise time. So resonant frequency and group delay calculations of pulses is very poor and affected by skin effects, even in controlled impedances, making Baseband communication much worse than the discrete equalized channel's of a modem for thruput in bps/Hz .....Tony
However hollow copper tubing with interior flash gold plating works wonders in microwave as does ENIG on stripline and gold-plated aluminum cases for RF circuits and enclosures for microwave. I saw this in '77.
Here is a different skin effect from UV on dielectrics (human skin) and how Sodium Bicarbonate helps prevent cancer. ( also reduce causes/reactions of itching). https://www.cancertutor.com/simoncini/. :):)