The cruical detail is buried in the post you link, where the author says:
"This renormalisation analysis that explained the phenomenon of “critical universality” was introduced by Kadanoff and Wilson in the 1960s, and is most powerful not in the real-space version introduced above, but in momentum space. (These techniques differ in important ways, but I can’t work out a way of explaining the differences concisely in this answer.)" [emphasis mine]
We're not "averaging fields over spacetime" when we do the actual momentum space version of renormalization. The infinities that renormalization is designed to counter in QFT come in two flavours - "infrared", because momenta go continuously down to zero and "ultraviolet" because momenta go arbitrarily high. The infrared issues are comparatively "easy" and rarely the focus of what we do because they can essentially be solved just by putting the theory into an arbitrarily large but finite box. The "ultraviolet" issues are the ones for which we really use renormalization.
The Wilsonian view is that we introduce a momentum cutoff $\Lambda$ above which some magic prevents momenta from existing. "Renormalizing" then means choosing a different cutoff $\Lambda'$. I guess you can view this as "averaging over" the momenta between $\Lambda'$ and $\Lambda$, but what we're averaging is the integrand of the path integral that gives us the partition function of our theory, not "the quantum fields". Explicitly, we view the partition function as a (Euclidean) path integral
$$ Z[J] = \int \mathrm{e}^{-L + J\phi}\mathcal{D}\phi$$
where $\mathcal{D}\phi = \Pi_{\lvert k\rvert < \Lambda} \mathrm{d}\phi_k$ with $\phi_k$ the Fourier modes of the fields (in a box) whose momentum is lower than our cutoff. Renormalizing to $\Lambda'$ means performing all the integrals over $\Lambda' < \lvert k \rvert < \Lambda$. This gives us some factor that we can write as $\mathrm{e}^{-L_\text{eff}}$ (this isn't trivial, but it's true at least perturbatively), and so the Lagrangian of our theory at $\Lambda$ effectively changes to some $L+L_\text{eff}$. The renormalization flow is now moving $\Lambda'$ around and looking at the different effective theories we get this way.
Generically, the effective Lagrangian contains all possible couplings of the fields to each other that aren't forbidden by a symmetry of the Lagrangian we started with, and so the "adjustable parameters" that the post you link talks about become a list of coupling constants - e.g. for $\phi^4$ theory, we get a coupling $\lambda_{2n}(\Lambda')$ for each even number of fields. The choice of $\lambda_{2n}(\Lambda')$ that produces the same scattering amplitudes as our original theory are the correct choice (because a theory whose scattering amplitudes differed would obviously be a different theory), so the "quantity invariant under renormalization" is "just" the S-matrix, not some high-level observable property. There are interesting observations to make that of all the possible $\lambda_{2n}(\Lambda')$, only a few - namely the "renormalizable" and "super-renormalizable" - are relevant at $\Lambda'\ll \Lambda$ (but this is another technical argument I don't want to add here), so if we suppose that the "natural" $\Lambda$ is very high, this can explain why only renormalizable QFTs seem to be relevant for describing our universe at the "low" energy scales $\Lambda'$ at which we're looking at it.
Lastly, let me remark that this Wilsonian view on renormalization is not universal - this is a nice conceptual view, but in practice "integrating out Fourier modes in the path integral" might be a bit of a hassle. Other renormalization schemes exist, and while they will also have a "cutoff" (more generically call "renormalization scale") $\Lambda$, their running couplings $\lambda_{2n}(\Lambda')$ will not agree with Wilsonian renormalization beyond leading order in perturbation theory.
