There is a lot to read about resistors and capacitors here:
Capacitor upgrade in crossover - Is it audible?
Part 2
Part 3
Differences between sand-cast and low-inductance resistors
Danny always refers to "smear" when looking at the coils, now this is not a term that is technically in use
The most important thing with core coils, as Danny also mentions, is the saturation limit. Once this is reached, the tonal effect is clearly audible.
If you stay well below the saturation limit of the core coil when listening to music, you will not suffer any disadvantages from using core coils, since the distortion produced is usually well below the distortion of the drivers.
Anyone listening at "normal" volume should not be able to drive a reasonably sized core coil into saturation.
I have described this in
more detail and with examples of core coil measurements here.
Within the different types of core coils there are still clear differences in quality - this is also mentioned in the video. So it really depends on the individual case.
But as
@amirm's distortion measurements of inexpensive speakers show us again and again, the distortions generated by the core coils tend not to play a major role - otherwise there would be no inexpensive speakers with low harmonic distortions.
What Danny doesn't directly address is the big advantage of using core coils - the significantly lower DC resistance (at a comparable price of core coil and air core coil), which leads to a noticeably higher sound pressure in the low frequency range (the area in which core coils are normally used).
If after replacing a core coil with an air core coil, a difference is perceived at normal listening volume, the difference in DC resistance is likely to be the cause - the air core coil reduces the sound pressure in the low frequency range somewhat (if the resistance of the air core coil is higher).