Let me phrase a proof of Serre's computation of the rational stable homotopy groups of spheres as stably as I can:

For every spectrum $X$, we can define its rationalization $X_{\mathbb{Q}}$ as the homotopy colimit $$\mathrm{hocolim}\, X \xrightarrow{2} X \xrightarrow{3} X \xrightarrow{4} \cdots .$$
As such a directed homotopy colimit translates into a usual colimit after applying homotopy groups, we obtain $\pi_* X_{\mathbb{Q}} \cong (\pi_* X) \otimes \mathbb{Q}$.

By a form of Hurewicz, we know that the negative homotopy groups of the sphere spectrum $\mathbb{S}$ are $0$ and its zeroth homotopy group is $\mathbb{Z}$. Thus, we obtain a Postnikov truncation map $\mathbb{S} \to H\mathbb{Z}$. After rationalization, this induces a map $\mathbb{S}_{\mathbb{Q}} \to H\mathbb{Q}$. We want to show that this map is an equivalence.

As these are bounded below spectra, Hurewicz/Whitehead II imply that we just need to check on homology. As homology also translates directed homotopy colimits into colimits, we obtain that $H_*(S_{\mathbb{Q}}; \mathbb{Z}) \cong \mathbb{Q}$ (concentrated in degree $0$). It suffices thus to show that the integral homology of $H\mathbb{Q}$ is the same (as the map is then automatically an isomorphism as it is one on $\pi_0$).

As the homology of $H\mathbb{Q}$ coincides with the homotopy of $H\mathbb{Q} \otimes H\mathbb{Z}$, we see that we can equally well compute the rational homology of $H\mathbb{Z}$. The $i$-th such homology is isomorphic to the colimit over $\widetilde{H}_{i+n}(K(\mathbb{Z}, n); \mathbb{Q})$. For $n$ odd, these groups are concentrated in degree $n$ and are $\mathbb{Q}$ there, implying the result. (This is the only place where I find it to be necessary to apply an unstable result.)

For the finite generation, I likewise need one input result, namely that the homology of $H\mathbb{Z}$ is degreewise finitely generated. Using unstable methods, this is again easy to prove as we obtain $H_i(H\mathbb{Z}; \mathbb{Z})$ as the colimit of $\widetilde{H}_{i+n}(K(\mathbb{Z}, n); \mathbb{Z})$. Each of the groups is finitely generated by an inductive argument using the Serre spectral sequence and the sequence stabilizes.

The argument is now the following: We prove simultaneously by induction that the $n$-th homotopy group of $\mathbb{S}$ is finitely generated and that the connective cover $\tau_{\geq n+1}\mathbb{S}$ has degreewise finitely generated homology. Hurewicz implies that $\pi_n \mathbb{S} \cong \pi_n\tau_{\geq n}\mathbb{S} \cong H_n(\tau_{\geq n}\mathbb{S}; \mathbb{Z})$. The latter group is finitely generated by assumption and hence also $\pi_n\mathbb{S}$. Thus $H\pi_n\mathbb{S}$ has degreewise finitely generated homology. Thus, the same is true for the fiber of $\tau_{\geq n}\mathbb{S} \to H\pi_n\mathbb{S}$, namely $\tau_{\geq n+1}\mathbb{S}$, which finishes the induction step.