What is the meaning of the expansion at first order ${\cal O}(\alpha_s)$ in $\delta_2$ and $\delta_3$ at the second step in the last line? These quantities are not "small" - on the contrary, the entire point is to then take the $\epsilon \to 0$ limit and the counterterms blow up.
The brief answer is that renormalization is first-and-foremost a perturbative formal power series in the coupling constant $\alpha_s$. E.g. a $Z$-factor is a formal power series $$Z~=~ \sum_{n=0}^{\infty} \alpha_s^nZ_n, \qquad Z_{n=0}~=~1.\tag{A}$$ Secondly, each coefficient $$Z_n=\sum_{m=-N}^{\infty}\epsilon^m Z_{nm}\tag{B}$$ of this formal power series is a truncated Laurent series in $\epsilon$. The coefficients are not necessarily small, as OP already has observed.
Eqs. (77)-(80) consider in particular the first-order coefficient $Z_{n=1}$.
See also e.g. this related Phys.SE post.
