The Zariski Riemann space, while an abandoned approach, has lead to later developments and generalizations, including $\text{Spv}$ (the space of valuations) and Huber's work. In studying it, I would like to know what fails if we erase the valuation ring assumption, and replace it with local ring.

Suppose we wanted to make a topological space on the local rings. Which desirable properties of the Zariski Riemann space fail when we try to make a topological space on the local rings contained in a field? For example, is it a spectral space?

Specifically, suppose we let $\text{Loc}(K)$ of a field $K$ be the topological space whose points are local rings, and whose open sets are unions of sets of the form $V(f_1, ..., f_n) = \{ R \subset K \text{ a local ring } : f_1, ..., f_n \in R \}$. This is completely analogous to the construction of the Zariski Riemann space.

One approach here would be the historical one: Zariski had Riemann surfaces in mind when creating the Riemann-Zariski space, and using local rings does not generalize this situation. Nevertheless, there is perhaps something more to say here about why it is better to use the Zariski-Riemann space instead of $\text{Loc}$.

We can make the Zariski Riemann space into a sheaf whose stalks at each point are the corresponding valuation rings. To do this, we create an étale topology on $\amalg_{R \in \text{Spv}(K)} R$ and let the sheaf $\mathcal{O}_{\text{Spv}(K)}$ be its sheaf of sections.

So we might try to make $\text{Loc}(K)$ into a sheaf whose stalks at each point are the corresponding local rings. To do this, define for each $a \in K$ a function $f_a : V(a) \rightarrow \amalg_{R \in \text{Loc}(K)} R$ sending $R \in V(a)$ to $(a, R) \in \amalg_{R \in \text{Loc}(K)} R$. We put a topology on $\amalg_{R \in \text{Loc}(K)} R$ where a sub-basis consists of images of functions of the form $f_{a}$ for $a \in K$. Then define $\mathcal{O}_{\text{Loc}(K)}$ to be the sheaf of sections of this étale cover.

We can then identify $\mathcal{O}_{\text{Loc}(K)} (U)$ with the intersection of local rings in $U$. The stalk $\mathcal{O}_{\text{Loc}(K), R}$ at $R$ is just $R$.

Does something go wrong here? What makes the Zariski-Riemann space more desirable topological space (or sheaf) to work with than $\text{Loc}(K)$?