First, let us assume that $p$ is odd for safety.

Then I think your impression is correct: the first statement of 12.5 (4), i.e the integrality of the module generated by the $\mathbf{z}^{(p)}_{\gamma}$, is true under the hypothesis that $T/pT$ is irreducible. Hypothesis (12.5.2) is used only to prove the second (and much more spectacular) statement, i.e the Iwasawa-theoretic divisibility.

When $E$ admits a $p$-isogeny, then C.Wuthrich has a result towards integrality. However, the published version is flawed and I admit I can't quite understand the correction on his webpage. Because C.Wuthrich is a frequent poster here, we can count on his explanation.

In general, proving that Euler systems are integral should be quite a difficult task because it is generally intimately linked with the question of whether the $\mu$-invariant vanishes.