What you're overlooking for the motivating situation (which should have been mentioned in the question as posed so that readers would have more context) is the final part of EGA III$_1$ 3.3.1. (Note that $Y'$ and $g$ there play no role whatsoever, by the way.)
In the setup of that result, set $\mathscr{S}$ to be the symmetric algebra over $\mathscr{O}_Y$ for the coherent sheaf $f_{\ast}(\mathscr{L}^{\otimes d})$ for an integer $d > 0$ (to be determined); this is graded quasi-coherent algebra of finite type on $Y$, generated in degree 1. Take $\mathscr{M}$ there to be $\mathscr{A} := \oplus_{n \ge 0} \mathscr{L}^{\otimes n}$ and set $p=0$. Observe that $\mathscr{M}$ is naturally a graded module over $f^{\ast}(\mathscr{S})$ by defining $\mathscr{M}_n = \oplus_{nd \le j < (n+1)d} \mathscr{L}^{\otimes j}$.
Hence, by (3.3.1.1) in loc. cit. if $\mathscr{M}$ is of finite type as an $f^{\ast}(\mathscr{S})$-module then there is an integer $k_0 \ge 0$ so that for $k\ge k_0$ we have $$\oplus_{(k+r)d \le j < (k+r+1)d} f_{\ast}(\mathscr{L}^{\otimes j})= \mathscr{S}_r \cdot \oplus_{kd \le j < (k+1)d} f_{\ast}(\mathscr{L}^{\otimes j})$$ for all $r \ge 0$. This means the natural multiplication map ${\rm{Sym}}^r(f_{\ast}(\mathscr{L}^{\otimes d})) \otimes f_{\ast}(\mathscr{L}^{\otimes j}) \rightarrow f_{\ast}(\mathscr{L}^{\otimes(rd+j)})$ is surjective for all $r \ge 0$ and all $j \ge k_0d$. It would then follow that the quasi-coherent $\mathscr{O}_Y$-algebra $f_{\ast}(\mathscr{A}) = \oplus_{n \ge 0} f_{\ast}(\mathscr{L}^{\otimes n})$ is generated by its (coherent) terms in degrees $\le k_0d$ as an $\mathscr{O}_Y$-algebra, so it would be of finite type as an $\mathscr{O}_Y$-algebra.
Thus, one just has to find $d>0$ for which the finite type graded $\mathscr{O}_X$-algebra $\oplus_{n \ge 0} \mathscr{L}^{\otimes n}$ is finite type as a graded module over the symmetric algebra on $f^{\ast}(f_{\ast}(\mathscr{L}^{\otimes d}))$. For this purpose it is certainly enough that $\mathscr{L}^{\otimes d}$ is generated by $f_{\ast}(\mathscr{L}^{\otimes d})$. In other words, it suffices that the natural map $f^{\ast}(f_{\ast}(\mathscr{L}^{\otimes d})) \rightarrow \mathscr{L}^{\otimes d}$ is surjective for some $d > 0$. That in turn is exactly one of the hypotheses in the motivating situation (which is presently not given in the question as posed).