The method for eliminating reflected current is borrowed from transmission lines. Somewhat like a transmission line, a large helical coil has a characteristic impedance in the range of a few thousand ohms. If we install a resistor from the top of the coil to ground and vary the resistance, we can find the characteristic impedance of the coil. When the impedance seen by the source at the bottom of the coil is equal to the value of the load resistance, we have found the characteristic impedance. Something akin to this exercise will yield the unknown characteristic impedance of a short piece of transmission line.
A piece of transmission line also has a velocity factor such that the actual velocity of an EM traveling wave through the transmission line is VF(c) where 'c' is the speed of light in a vacuum. In like manner, a loading coil has a velocity factor. If we can suceed in eliminating reflected current and thus measure the phase shift through a loading coil, we can calculate the velocity factor which is typically in the range of 0.01-0.03 for mobile loading coils. The speed of an EM wave through a typical loading coil is slowed by a ballpark factor of roughly 50 to one compared to a vacuum.
Can EZNEC be used to shed some light on the "current through a loading coil" controversy? EZNEC faithfully reports the standing wave current in a standing wave antenna and faithfully reports the traveling wave current in a traveling wave antenna. The goal of this exercise for EZNEC used with a mobile loading coil, is to eliminate the reflected current through the coil so EZNEC will display the forward traveling wave current rather than the standing-wave current. Using that technique, the phase shift (delay) through a loading coil can be readily observed using the EZNEC "loads" feature.
A large loading coil, like a 75m Texas Bugcatcher loading coil, has a characteristic impedance (like a transmission line) and a velocity factor (like a transmission line). An IEEE white paper  indicates that a 75m Texas Bugcatcher coil should have a characteristic impedance of a few thousand ohms and a velocity factor between ~0.01 and ~0.03. If we treat the 75m Bugcatcher loading coil as a transmission line and load it with a resistance equal to its characteristic impedance, we can minimize (if not eliminate) the reflected current leaving only the forward traveling current. Let's see if we can use EZNEC to accomplish that task.
If we load the coil with its characteristic impedance, reflected current is eliminated (or at least minimized). The Z0 of the Texas Bugcatcher coil is apparently ~2745 ohms at 7.96 MHz and ~1975 ohms at 3.8 MHz. The VF (velocity factor) is apparently ~0.016 on 7.96 MHz and ~0.018 on 3.8 MHz. The coil is six inches in diameter, 7.5 inches long, and is 1/4WL self-resonant on 7.96 MHz. VF = 0.5'/(246'/7.96 MHz) = 0.01618
Here is the graph of the currents through the coil as reported by EZNEC:
Now let's make the following assumptions:
1. Assume there is no current amplitude "drop" through the coil for the forward current or for the reflected current. The amplitude of the forward current is the same at both ends of the coil. The amplitude of the reflected current is the same at both ends of the coil. The coil has been modeled as lossless so this assumption seems fair.
2. Assume that the amplitude of the reflected current is equal to the amplitude of the forward current. This seems like a fair assumption in a lossless system and works for lossless stubs.
3. Since the forward current and reflected current are in phase at the base of the coil and the phasors are rotating in opposite directions, assume the reflected current phase is the mirror image of the forward current phase. This seems to be an accurate assumption since these two phasors are reported by EZNEC and Kraus to sum to very close to zero phase.
4. Assume 0.5 amps of forward current flowing into the bottom of the coil and out of the top of the coil. Assume 0.5 amps of reflected current flowing into the top of the coil and out of the bottom of the coil.
The following graphic assumes all of the above.
Remembering that there is zero loss in our coil, let's phasor-add the forward current and the reflected current. The result is the following graphic:
This last graph is pretty close to measurements and EZNEC simulations. In fact, if we take the coil, remove the load resistor, and install a nine foot whip, we will bring the system to resonance. EZNEC reports that the graph of the current in that case is identical to Figure 4. That EZNEC file can be downloaded at: zipped coil505t.EZ Apparently, the primary factor in the net current at the bottom and top of a loading coil is the superposition of the forward and reflected currents, not the losses in the coil, not the radiation from the coil, and not even the local environment of the coil. The effects of these last factors appear to be secondary compared to the superposition of the forward and reflected currents.
Note that the measured net current amplitude "drop" through the coil is an illusion caused by the phasing of the forward and reflected currents. There is no drop in either of these component currents, i.e. they are both of the same amplitude at each end of the coil.