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OK, after my last reply specifically about the AMT3000, here are some general comments relevant to Radio Joe’s original post.
Connecting the coax to the top of the loading coil where the antenna is connected will not work. Assuming you have a transmitter with a 50 ohm output impedance feeding 50 ohm coax, you will basically “short out” the loading coil. Ernie Ross said this in different words. The voltage across 50 ohms with 100 mW power is 2.24 VRMS. Typical voltage at the top of a loading coil at 100 mW is about 140 VRMS or more for a fairly efficient transmitter and coil.
The problem with base-loaded antennas is that the resonant resistance at the bottom of the coil is almost totally dependent on the ground system. The coil may introduce maybe 10 ohms, but the ground can range from an ideal of zero up to over 100 ohms, so the resistive impedance at the bottom of the coil would range from 10 ohms to 110 ohms or more. A 50 ohm feed at the bottom of the coil will be a mismatch to a varying extent. This is one reason not to use coax (aside from FCC rules).
The best overall transmitter/antenna arrangement for highest efficiency is to connect the output of the transmitter directly to the bottom of the loading coil with a short piece of wire (no coax) and to have a transmitter that has a means for adjusting the loading to match the actual antenna load (depends on ground quality).
If the transmitter load impedance is specified at say 50 ohms, then you can compensate for the lower than 50 ohm load resistance of a very well grounded antenna by feeding the coil a few turns up from the bottom and grounding the bottom of the coil. But if your ground resistance + coil loss is higher than 50 ohms, you have no option other than to improve your ground to get the total load down to 50 ohms.