An explanation for this?

[mike3]

And also, why this is not the accepted extension for tetration of rational values?

Because it is not theoretically safe.
There is no proof that the Dmitrii's (and also Sheldon's) procedure of computing the sexp even converges.
There is however a theoretic uniqueness criterion and a proven procedure that exactly produces the corresponding sexp/slog. I found that out recently and give links/references perhaps later. However Dmitrii's and also Sheldon's way is simpler to calculate and seem to compute exactly this unique sexp/slog.

What is this proven algorithm, and does it provide any clues to finding the explicit (or close enough, e.g. if in terms of, say, infinite sums) coefficients of the Taylor series for the tetrational function, i.e. what is \( a_n \) in

\( \mathrm{tet}(z) = \sum_{n=0}^{\infty} a_n z^n \)

?