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I can’t answer all of your questions but here is some information I have that may get you started understanding the situation.
I built an antenna similar to the one on the SSTRAN site. The difference is that I used #14 wire on the same form. To get the required L, I had to “wind back” a second layer from the top of the form. I mounted the antenna and coil over 10 10′ radials layed on top of the earth. I tuned the antenna for best response at 1520 kHz. The inductance at the tap which produced maximum test point reading in the SSTRAN was at an inductance of 250 uH. (measured with the coil removed from the antenna/ground between the tap and the top of the coil with a HP4250A bridge).
Many view the coil function as cancelling the reactance of the radiator. There is another way to look at the interaction between C5/C23 and the antenna and coil. C5/C23 and the coil form a L network and if you include the radiator capacitance, they form a pi network. The antenna C does not appear in parallel with C5/C23 since the coil is between them. L and pi networks are commonly used for impedance transformation in radio transmitters. It is likely, though I have not analysed this for the SSTRAN, that the combination of C5/C23 and the loading coil form a L network to transform the highly reactive impedance of the radiator to one which is acceptable to the transmitter and that adjustment of C5 optimizes this transformation for the transmitter. L4 thru L7 probably serve the same function when using a wire antenna. If this is the case, I see no way to simplify the design, but perhaps you will fine one.