Schemes as categories fibered in thin groupoids Every time I start to read about schemes from a birds-eye view (like in the introduction to The Geometry of Schemes by Eisenbud and Harris) I get really excited; they sound like a categorical approach to geometry, allowing for a rigorously formalized account of the infinitesimal which almost directly permits one to formalize their intuition about infinitesimal geometric concepts and then reason about these concepts precisely in an intuitively satisfying manner.
By the time I get to the formal introduction, however, we're immediately talking about prime ideals of commutative rings and topological spaces -- I have no problem with these concepts in a vacuum, but this approach feels far off base from the expectations outlined above. I have no doubt that category theory is hiding nearby in the background, but  the approach through commutative algebra and topology is off-putting to me.
In the midst of my despair I came across Martin Brandenburg's answer to a question over at MSE where he says that 'schemes are categories fibered in setoids', however the page of the Stacks project linked in the answer makes no reference to schemes. In the comments on this same answer he outlines how the category of setoids is isomorphic to the category of thin groupoids, which leads to the following

Proposition. A scheme is a category fibered in thin groupoids; that is, a Grothendieck fibration $p:\mathcal{E}\to\mathcal{B}$ whose fibres are all thin groupoids.

Since Grothendieck fibrations aren't the fibrations in the model structure on $\mathfrak{Cat}$, we may want instead to offer the following

Proposition. A scheme is a category essentially fibered in thin groupoids; that is, a Street fibration $p:\mathcal{E}\to\mathcal{B}$ whose essential fibres are all thin groupoids.

Either of these points of view would be very appealing to me.

Question: Are either of these propositions correct, and if so are there any introductions to the theory of schemes viewing them in this light?

 A: This is essentially the functor of points approach to schemes.
Define the site AffSch of affine schemes as the opposite
category of commutative rings, equipped with the Zariski Grothendieck topology.
Concretely, the poset (locale) of opens of the Zariski spectrum
of a commutative ring R can be identified with the poset of radical ideals of R; open covers are given by collections of radical ideals in R that generate an ideal whose radical equals R.
Consider the category S of sheaves of sets on the site AffSch.
Various objects considered in algebraic geometry, such as schemes and algebraic spaces, form full subcategories of S.
Concretely, schemes can be characterized as objects of S that admit an atlas,
i.e., a family of open immersions from affine schemes such that the induced map from their coproduct is an epimorphism.
The nLab has another characterization of schemes, see Definition 2.4 there.
The 2-categories of thin groupoids and sets (the latter with identity 2-morphisms) are equivalent.
Thus, sheaves of sets on AffSch (equivalently, categories fibered in sets over AffSch) can be replaced with the equivalent 2-category of categories fibered in thin groupoids over AffSch.
The cited answer by Martin Brandenburg
gives an example when such an adjustment is (marginally) useful,
since it allows us not to take isomorphism classes for certain constructions.
A lot of books cover the functor of points approach in some way, including Vakil's modern exposition.
Among the more classical sources one can point to Demazure and Gabriel's Introduction to Algebraic Geometry and Algebraic Groups (North-Holland, 1980).
