The question stems with a misinterpretation of Theorem 1.1 in the paper by Boileau and Zieschang. Theorem 1.1 excludes a fair number of cases, in particular, it does not apply to (totally oriented) closed Seifert manifolds with 3 singular fibers and base of genus 0. Some of these excluded Seifert manifolds provide counter-examples to your claim about rank $\ge 3$.

For instance, take $N$ which is the exterior of a $(p,q)$-torus knot which is nontrivial and not the trefoil. Then the genus of this knot is $$ g=\frac{(p-1)(q-1)}{2}\ge 2 $$ (because I excluded the trefoil which has genus 1). The manifold $N$ is a surface bundle over the circle whose fiber $F$ is the once-punctured surface of genus $g$. The monodromy of this fibration is a finite order (actually, the order is $pq$) homeomorphism $h: F\to F$. Thus, if we collapse the boundary of $F$ to point, we obtain a closed surface $S$ of genus $g$ and $h$ will project to a finite order homeomorphism $f: S\to S$. The mapping torus $M=M_f$ is a Seifert manifold of type ${\mathbb H}^2\times {\mathbb R}$ obtained by a Dehn filling of the boundary of $N$. The base of the Seifert fibration will have three singular points and genus 0: Two of the singular fibers come from $N$ and one comes from the solid torus attached to $\partial N$ as the result of our Dehn filling. (It is a general fact that the mapping torus of a finite order homeomorphism of a hyperbolic surface is a Seifert manifold of type ${\mathbb H}^2\times {\mathbb R}$.) Since the group $\pi_1(N)$ is 2-generated, the quotient group $\pi_1(M)$ is also 2-generated.

The question stems from a misinterpretation of Theorem 1.1 in the paper by Boileau and Zieschang. Theorem 1.1 excludes a fair number of cases, in particular, it does not apply to (totally oriented) closed Seifert manifolds with 3 singular fibers and base of genus 0. Some of these excluded Seifert manifolds provide counter-examples to your claim about rank $\ge 3$.

For instance, take the exterior $N$ of a $(p,q)$-torus knot which is nontrivial and not the trefoil. The genus of this knot is $$ g=\frac{(p-1)(q-1)}{2}\ge 2 $$ (because I excluded the trefoil which has genus 1). The manifold $N$ is a surface bundle over the circle whose fiber $F$ is the once-punctured surface of genus $g$. The monodromy of this fibration is a finite order (actually, the order is $pq$) homeomorphism $h: F\to F$. Thus, if we collapse the boundary of $F$ to point, we obtain a closed surface $S$ of genus $g$ and $h$ will project to a finite order homeomorphism $f: S\to S$. The mapping torus $M=M_f$ is a Seifert manifold of type ${\mathbb H}^2\times {\mathbb R}$ obtained by a Dehn filling of the boundary of $N$. The base of the Seifert fibration will have three singular points and genus 0: Two of the singular fibers come from $N$ and one comes from the solid torus attached to $\partial N$ as the result of our Dehn filling. (It is a general fact that the mapping torus of a finite order homeomorphism of a hyperbolic surface is a Seifert manifold of type ${\mathbb H}^2\times {\mathbb R}$.) Since the group $\pi_1(N)$ is 2-generated, the quotient group $\pi_1(M)$ is also 2-generated.