Skin effect is, indeed, not a problem at DC.
However, at AC frequencies up to about 1 KHz, the impedance of wire decreases, it doesn't increase. The governing equation for Z_0 for AC below 1KHz is sqrt(R/(j2(pi)fc)). Above about 100 KHz it becomes sqrt(L/C), and R doesn't even figure into the equation. From about 1 KHz to 100 KHz, skin effect starts to take effect.
This is because skin effect, the confining of electrons to an area near the surface of a solid circular-cross-section conductor, is frequency dependent, not just "it's AC, there's skin effect". In short, the skin depth at low frequencies is significantly larger than the radius of the wire, and no significant confining is happening.
Between 1 KHz and 100 KHz (audio and ultra-sound frequencies) the crossover between raw resistance, decreasing because f in the denominator of the fraction under the radical is increasing, begins to lose out to skin effect, which reduces the cross-sectional area of affected free electrons. At 100 KHz, the entire motion of electrons is confined to the surface, and tubes become more cost-effective and efficient for carrying the signals.
Below 1 KHz, the impedance of a 75ohm video cable can rise to as much as 4000ohms, dependent on frequency due to capacitive effects. Plain wire has considerably capacitive effect at low frequencies (unless it is poorly routed). The increased impedance is not "based on skin effect", though: it is entirely unaffected by skin effect. At 400 Hz, though, the impedance of a video cable still makes it unattractive to use for routing control signals, but it is still not an affect of the skin effect, which has not begun to reduce cross-sectional area of motile electrons.
1 - At very high frequencies, the skin effect is so pronounced that current exits only over a very thin layer of good (metallic) conductor.
2 - The penetration of current in a good conductor is characterized by the skin depth.
3 - The skin depth is inversely proportional to the square root of frequency, permeability, and conductivity.
4 - Here an example of skin depth for copper and iron:
Skin depth for copper at 60Hz (8.6mm),1 KHz (2.1mm), 1 MHz (0.067mm).
Skin depth for iron at: 60Hz (0.6 mm), 1 KHz (0.16mm), 1 MHz (0.0053 mm).
Skin effect has no bearing on the subject of control (or audio electronics, either). Let's dispense with the myth.