I know very little about the absolute/algebraic geometry side, but I think I understand the gist of the category theory going on here. I guess this answer might require one to know a bit of both the stable homotopy story and the $\mathbb{F}_1$-geometry story.
tl;dr is that, no, algebra over $\mathbb{S}$ requires that the underlying objects be spectra, which are a derived version of abelian groups, in the sense that "discrete $\mathbb{S}$-modules" are exactly abelian groups; but algebra over $\mathfrak{s}$ is (i) entirely discrete or $1$-categorical and not derived at all (i.e. there's no notion of weak equivalence at all, only of isomorphism) and (ii) even if we took $\mathfrak{s}$-modules to be the discrete version of some framework for derived algebra, it wouldn't be $\mathbb{S}$-modules because $\mathfrak{s}$-modules contain things like hypergroups, tropical rings, and commutative monoids, so their "derived theory" will need to be based in $(\infty,n)$-categories rather than $(\infty,1)$-categories, as $\mathbb{S}$-modules is.
I'll try to follow Connes and Consani and say $\mathfrak{s}$-modules instead of $\Gamma$-sets, but I'll almost certainly just start using them interchangeably at some point. So anyway, we've got $\mathfrak{s}$-modules and we've got $\mathbb{S}$-modules. First let's clarify the difference between $\mathfrak{s}$ and $\mathbb{S}$, as, in a sense, they are both the sphere spectrum, but I think this point of view is misleading. What we're calling $\mathfrak{s}$ here is the inclusion $\Gamma\hookrightarrow Set_\ast$, whereas, for the time being at least, $\mathbb{S}$ is the unit of the $\infty$-category of spectra, if you like, or the suspension spectrum of $S^0$.
Recall that there is a model structure on $\Gamma$-spaces, called the stable model structure, whose fibrant objects are the very special $\Gamma$-spaces. Namely, those that satisfy the Segal condition, roughly $X[n_+]\simeq X[1_+]^n$ where the right hand side has $n$ factors, and such that $X[1_+]$ is an abelian group (here I'm using $n_+$ to denote the set with $n+1$ elements, one of which is the basepoint). These model "commutative topological groups up to homotopy," and therefore grouplike $\mathbb{E}_\infty$-spaces, and therefore connective spectra. Now, notice that the inclusion $\Gamma\to FinSet_\ast\hookrightarrow Set_\ast\hookrightarrow sSet_\ast$ does not define a special $\Gamma$-space. To see this, consider $1_+\times 1_+=\{\ast,1\}\times\{\ast,1\}=\{(\ast,\ast),(\ast,1),(1,\ast),(1,1)\}\cong3_+\neq 2_+$. So $\mathfrak{s}$ itself is not fibrant in $\Gamma$-spaces with the stable model structure and therefore doesn't correspond to any connective spectrum. Of course the way that we get around this in model categories is that we fibrantly replace. It's a theorem that when we fibrantly replace $\mathfrak{s}$ (thought of as a discrete $\Gamma$-space) we get $\mathbb{S}$.
There's a concrete way to describe fibrant replacement in this category, which is to formally invert the so-called Segal maps, i.e. localize at them, and group complete $X[1_+]$ (thanks to Chris Schommer-Pries for explaining this part to me). In a very real way this deserves to be called "group completion." The first part is, basically, "commutative monoid completion" since it forces the multiplication to be defined everywhere and to be singly defined (an arbitrary $\Gamma$-space can be thought of as a sort of $\mathbb{E}_\infty$-monoid object in which the multiplication is either multi-valued or only partially defined). The second part is formally adding inverses. The second part is what we usually think of as group completion of course, but if you're starting with something that's not even a commutative monoid, you've got to do a bit more.
So, if we believe that "fibrant replacement in the stable model structure" can reasonably be called group completion, then that's exactly what $\mathbb{S}$ is, it's the group completion of $\mathfrak{s}$ (or, if you accept Connes and Consani's contention that $\mathfrak{s}=\mathbb{F}_1$, then $\mathbb{S}$ is the group completion of $\mathbb{F}_1$, which is consistent with interpreting the Barratt-Priddy-Quillen as saying $K(\mathbb{F}_1)\simeq \mathbb{S}$; notice that although Barratt-Priddy-Quillen says that $K(FinSet)\simeq \mathbb{S}$, it's still true that $K(\mathfrak{s}-Mod)\simeq \mathbb{S}$).
The category of $\Gamma$-spaces, or $\Gamma$-sets, has a symmetric monoidal structure coming from Day convolution. In this monoidal structure, $\mathfrak{s}$ is the monoidal unit. So everything in $\Gamma$-set deserves to be called an $\mathfrak{s}$-module, or an $\mathbb{F}_1$-module. But now, given an $\mathfrak{s}$-module $M$, one can further say that $M$ itself is a monoid with respect to the Day convolution monoidal structure, and this is what Connes and Consani call an $\mathfrak{s}$-algebra. The key insight here for them is that most of the existing models of "$\mathbb{F}_1$-algebra" are subsumed by this construction. In particular, commutative monoids give $\mathfrak{s}$-modules and semirings give $\mathfrak{s}$-algebras; a large class of abelian hypergroups and hyperrings embed into $\mathfrak{s}$-modules and $\mathfrak{s}$-algebras (for the two preceding examples we're using this fact that $\Gamma$-sets can be thought of as having either a partially defined or multivalued abelian group structure); and perhaps most excitingly, Durov's algebraic monads embed into $\Gamma$-sets, so we get things like tropical rings, and these generalized rings that Durov uses to do things like Arakelov geometry (Connes and Consani have also written about a connection between their $\mathfrak{s}$-modules and Borger's $\Lambda$-ring model for $\mathbb{F}_1$-geometry, but I haven't tried to understand that; it's also not at all clear to me, or anyone else as far as I know, that Lorscheid's blueprint stuff, and all the things related to it, can be described in terms of Connes and Consani's work).
I should mention that one can take Durov's model of $\mathbb{F}_1$ and map it (via the so-called assembly map of Lydakis that Connes and Consani use) to $\Gamma$-sets, and one does, as an object, get $\mathfrak{s}$ (I think). But, if I'm not mistaken, the modules over Durov's $\mathbb{F}_1$, in Durov's setup, are less general than $\mathfrak{s}$-modules.
One can of course talk about "simplicial $\mathfrak{s}$-modules," but if we think about what a simplicial $\mathfrak{s}$-module is going to be, well it's the same as a $\Gamma$-space (where by space I mean simplicial set) so if one wishes to do homotopy theory in the usual way, I think one is still going to end up back at something like $\mathbb{E}_\infty$-spaces, which loses all the examples that Connes and Consani are interested in (for instance, they seem pretty interested in this adèle class space construction, which gives a hyperring, so gets destroyed if you force the addition to be single-valued).
Moreover, recall that if we have an $\mathbb{E}_\infty$-space $X$ then we get the associated $\Omega$-spectrum, i.e. connective $\mathbb{S}$-module, by taking the infinite delooping $HX=\{X,BX,B^2X,B^3X,\ldots\}$. This doesn't even make sense if $X$ isn't a very special $\Gamma$-space. So to do "homotopy theory over $\mathbb{F}_1$," we have to be a bit more clever. I.e., prima facie, "derived $\mathbb{F}_1$-algebra" or "derived $\mathbb{F}_1$-geometry" is going to be very different from spectral algebra(ic geometry).
I think the way to get to derived $\mathbb{F}_1$-algebra, and have it be more general than spectral algebra (i.e. avoiding group completion), is that we're going to have to work with something like complicial $\Gamma$-sets, rather than simplicial. The idea here is that simplicial sets are pretty good at modeling $(\infty,1)$-categories, but pretty bad at modeling $(\infty,n)$-categories, and if you want to take the infinite delooping of, say, a commutative monoid $M$, you're going to need $B^nM$ to be an $n$-category of some sort, because you're going to have one object with an $M$'s worth of non-invertible $n$-cells, and complicial sets seem like a decent way to model categories like this (although there are other options too of course, and I haven't worked out any of the details at all). But whatever your category of $\mathbb{F}_1$-spectra is, there should be a group completion functor landing in $\mathbb{S}$-modules.
So if we're following Connes and Consani's proposed path to a proof of the Riemann Hypothesis, we can't work over $\mathbb{S}$, because all of the machinery that Connes and Consani want to use disappears (because we're group completing). In particular, with respect to your plan of taking an affine scheme over $\mathbb{F}_1$ and producing the associated spectral scheme, what you're basically doing is group completing an $\mathbb{F}_1$-algebra to some commutative connective ring spectrum $A$, then taking spec of that. But, again, this is destroying most of the tools Connes and Consani are using or are interested in.
If you wanted to start from scratch and say "Let's figure out how to rewrite Deligne's proof of the function fields Riemann Hypothesis by thinking of $H\mathbb{Z}$ as a curve over $\mathbb{S}$," well, I think you're going to struggle. I certainly wouldn't go on the record as saying it's impossible, but the sorts of machinery you'd want to use either don't exist over $\mathbb{S}$ or, at the time being at least, don't make sense (cf. Tyler Lawson's answer here: Dedekind spectra). Although you should take this last assertion of mine with a large grain of salt. Might make sense to ask Lars Hesselholt or Bjørn Dundas or somebody about this.
