(Rhetorical Question)
Given that the coil section resembles a high-Zo transmission line with
reflections, how can the current into the coil section be equal in
magnitude and phase to the current out of the coil section?
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(Rhetorical Answer)
It can't. Are you trying to insinuate I've ever said that it could?

But I'm pleased to see you are getting nearer to understanding how to model
a loading coil in an antenna wire. After all, the concept is simplicitly
itself to any electrical engineer worth his salt. And it's intuitively
obvious it's the only way of going about the job. Your 'cosine formula'
automatically comes out in the wash. As Holmes said to Watson. ;o)

Incidentally, from memory, in the 1950's, in the IEEE "Reference Data for
Radio Engineers" there was an article on short helically-wound antennas,
giving number of turns for a given height of 1/4-wave resonance. The author
was on the right track. But there were errors in the formulae which had not
been arrived at by analytical means.
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Reg, G4FGQ