Let $\mathcal{C}$ be a small site (or, at least, cofinally small) and let $\textbf{Psh} (\mathcal{C})$ be the category of presheaves on $\mathcal{C}$. There is a functor $\Gamma : \textbf{Psh} (\mathcal{C}) \to \textbf{Set}$ represented by the terminal presheaf (which may or may not be representable in $\mathcal{C}$, at this level of generality), and it has a left adjoint $\Delta : \textbf{Set} \to \textbf{Psh} (\mathcal{C})$ that sends every set $A$ to the "constant" presheaf defined by $(\Delta A) (U) = A$. We have a counit morphism $\epsilon_F : \Delta \Gamma F \to F$ for every presheaf $F$, and your construction is precisely the image of this morphism. Expressed this way, the failure of $\operatorname{Im} \epsilon_F \subseteq F$ to be a sheaf becomes unsurprising: usually we have to sheafify the presheaf image to obtain a sheaf.
If your goal is to construct a flabby (pre)sheaf, then it would be inappropriate to sheafify $\operatorname{Im} \epsilon_F$: sheafification can destroy flabbiness. On the other hand, if we work with presheaves then representability is a rather strong condition: indeed, representable presheaves are projective, so the epimorphism $\Delta \Gamma F \to \operatorname{Im} \epsilon_F$ would be split. But that would make $\operatorname{Im} \epsilon_F$ a retract of a constant presheaf, hence also constant – not very interesting, I think.
Finally, let me remark that the notion of flabby (pre)sheaf does not seem to be appropriate for non-localic sites. The point of flabby sheaves of modules on a topological space or locale is that they are acyclic with respect to the global sections functor, but I don't think this is true for a general site.
If your goal is to read Demazure and Gabriel's book, then (as I explained in the comments) your question is based on false premises and the solution is to read the definitions carefully.
But let me address your question as written, since it will illuminate why Demazure and Gabriel use the definitions they do.
First, as you observe, the category of elements of an arbitrary functor $F : \textbf{CRing} \to \textbf{Set}$ is not always small.
Actually, it is almost never small, because $\textbf{CRing}$ itself is not small: as soon as $F (A)$ is non-empty for all rings $A$, then the category of elements of $F$ will be at least as big as $\textbf{CRing}$.
This is not actually fatal for the problem at hand (though it does introduce many complications).
It sometimes happen that the functor $F$ you are interested in is a colimit of a small diagram of representable functors, i.e. there is a small diagram $A : \mathcal{I}^\textrm{op} \to \textbf{CRing}$ such that $F (B) \cong \varinjlim_\mathcal{I} \textbf{CRing} (A, B)$ naturally in $B$.
In that case, you can compute the colimit $\left| F \right|$ you seek as $\varinjlim_\mathcal{I} \operatorname{Spec} A$.
The biggest complication is that $\left| F \right|$ is not well defined for arbitrary $F$, so you only get a partially defined functor $\left| - \right|$.
If you try to restrict to a full subcategory of functors $\textbf{CRing} \to \textbf{Set}$ on which $\left| - \right|$ is well defined everywhere, you then have the complication that the putative right adjoint may not have image contained in that subcategory.
I am not aware of any good way to resolve this dilemma; I think you have no choice but to settle for a partially defined adjoint.
Now for some good news: there is a clean necessary and sufficient condition for a functor $F : \textbf{CRing} \to \textbf{Set}$ to be a colimit of a small diagram of representable functors.
Definition.
Let $\kappa$ be an infinite regular cardinal.
A $\kappa$-accessible functor is a functor that preserves $\kappa$-filtered colimits.
Proposition.
Let $F : \textbf{CRing} \to \textbf{Set}$ be a functor.
The following are equivalent:
- $F$ is $\kappa$-accessible.
- $F$ is the left Kan extension of a functor $\textbf{CRing}_\kappa \to \textbf{Set}$ along the inclusion $\textbf{CRing}_\kappa \hookrightarrow \textbf{CRing}$, where $\textbf{CRing}_\kappa$ is the full subcategory of $\kappa$-presentable rings (i.e. rings presentable by $< \kappa$ generators and $< \kappa$ relations).
- There is a small diagram $A : \mathcal{I}^\textrm{op} \to \textbf{CRing}$ such that $F \cong \varinjlim_\mathcal{I} \textbf{CRing} (A, -)$ and, for each $i$ in $\mathcal{I}$, $A (i)$ is a $\kappa$-presentable ring.
The functor $R \mapsto R^{\oplus \mathbb{N}}$ you mention is easily seen to preserve filtered colimits (i.e. be an $\aleph_0$-accessible functor).
It is just as easy to see that it is the colimit of a small (indeed, countable!) diagram of representable functors, namely,
$$\textbf{CRing} (\mathbb{Z}, -) \longrightarrow \textbf{CRing} (\mathbb{Z} [x_1], -) \longrightarrow \textbf{CRing} (\mathbb{Z} [x_1, x_2], -) \longrightarrow \cdots$$
where the maps are the ones induced by the homomorphisms $\mathbb{Z} [x_1, \ldots, x_n, x_{n+1}] \to \mathbb{Z} [x_1, \ldots, x_n]$ that send $x_i$ to $x_i$ for $1 \le i \le n$ and $x_{n+1}$ to $0$.
Thus, the geometric realisation of $R \mapsto R^{\oplus \mathbb{N}}$ is the colimit $\varinjlim_n \mathbb{A}^n$.
I suppose I owe you an example of a functor $\textbf{CRing} \to \textbf{Set}$ that is not accessible.
Choose an ordinal-indexed sequence of fields, $K_\alpha$, such that $K_\alpha$ is strictly smaller in cardinality than $K_\beta$ whenever $\alpha < \beta$.
Let $F (R) = \coprod_{\alpha} \textbf{CRing} (K_\alpha, R)$ for non-zero rings $R$ and let $F (\{ 0 \}) = 1$.
Since any ring homomorphism $K_\alpha \to R$ is injective when $R$ is non-zero, $\textbf{CRing} (K_\alpha, R)$ is empty for sufficiently large $\alpha$, so $F (R)$ is indeed a set.
On the other hand, it is clear that $F$ cannot be the left Kan extension of any functor $\textbf{CRing}_\kappa \to \textbf{Set}$: if it were, it would be impossible to distinguish between this $F$ and the one where we cut off the disjoint union at some ordinal $\beta$ such that $K_\beta$ is not $\kappa$-presentable.
Best Answer
The answer is "yes"! The closure of a subfunctor is defined to be the intersection of all closed subfunctors containing it. See, for example, Demazure and Gabriel's excellent book: "Introduction to Algebraic Geometry and Algebraic Groups".