It seems you actually understand the situation very well!
Borel's definition is a hybrid between classical algebraic geometry and scheme theory.
It stems from the desire not to use the full machinery of schemes.
Technically Borel can get away with that approach because for a scheme $X$ of finite type over a field $K$, the subset of closed points $X_{cl} \subset X$ is very dense in $X$.
This means that the restriction map $Open(X) \to Open (X_{cl}): U \mapsto U\cap X_{cl}$ is bijective.
The reason for that is that a finitely generated algebra $A$ over $K$ is a Jacobson ring, meaning that every prime ideal in $A$ is the intersection of the maximal ideals which contain it.
And for Jacobson rings we actually have functoriality: given a morphism $A\to B$ between two Jacobson rings, the inverse image of a maximal ideal of $B$ is a maximal ideal of $A$.
But I feel that this ad hoc approach should be a temporary crutch. The sooner you handle full-fledged scheme theory, the better: I strongly encourage you to go on reading Hartshorne!
I want to respond to one particular statement you made, but first I'll make things a bit more precise.
We have the adjunction $$\newcommand\Hom{\operatorname{Hom}}\newcommand\Spec{\operatorname{Spec}}\newcommand\calO{\mathcal{O}}\Hom(X,\Spec A)\simeq \Hom(A,\Gamma(X,\calO_X)),$$ and in particular this tells us that
$$\Hom(X,\Spec \Gamma(X,\calO_X))\simeq \Hom(\Gamma(X,\calO_X),\Gamma(X,\calO_X)).
$$
The identity map $\Gamma(X,\calO_X)\to\Gamma(X,\calO_X)$ therefore gives us a map of schemes $X\to \Gamma(X,\calO_X)$ as you noticed, and this is indeed the canonical map. However, you seem to be under the impression that this map is therefore an isomorphism.
In general this cannot possibly be true, since if the map were an isomorphism, $X$ would necessarily have to be affine. However, if $X$ is affine, this map is indeed an isomorphism.
Let's be a little more clear how this map works then.
In fact let's be a little more clear how it works in general. Let $\phi : A\to \Gamma(X,\calO_X)$ be a ring morphism. Let's try to understand the induced map $f : X\to \Spec A$.
Let $U$ be an affine open in $X$. Then we have the maps
$$\newcommand\toby\xrightarrow A\toby{\phi}\calO_X(X)\toby{r_{XU}} \calO_X(U).$$
Taking $\Spec$ of this sequence gives
$$U\toby{\Spec r_{XU}} \Spec \calO_X(X) \toby{\Spec \phi} \Spec A.$$
Gluing these maps together gives the desired map from $X$ to $\Spec A$.
Observe then that if $\phi=\newcommand\id{\operatorname{id}}\id$, that the map $X\to \Spec \Gamma(X,\calO_X)$ is the result of gluing the maps obtained from applying the Spec functor to the restrictions $r_{XU}:\calO_X(X)\to \calO_X(U)$.
If $X$ is affine, then we can take $U=X$, and there's no need to glue, the map $X\to \Spec\Gamma(X,\calO_X)$ is $\Spec \id=\id$. On the other hand, if $X$ is not affine, for example, if $X$ is a projective $k$-scheme with $k$ algebraically closed, then $\Gamma(X,\calO_X)=k$, and $X\to \Spec\Gamma(X,\calO_X)$ is the $k$-scheme structure morphism $X\to \Spec k$, which in general is clearly not an isomorphism.
Best Answer
Here's one motivation. Suppose you want to do "geometry." By this I mean you are working with varieties over an algebraically closed field, say $\overline{\mathbb{Q}}$. By working in the category of schemes over $S=Spec(\overline{\mathbb{Q}})$ you kill off all sorts of morphisms that aren't "coming from geometry." We'll get to what that means after an example.
If we are honestly doing geometry, then we would expect that since $S$ is just geometrically a single point that it's automorphism group would be trivial. But consider any $\sigma\in Gal(\overline{\mathbb{Q}}/\mathbb{Q})$. As long as $\sigma$ is not the identity map, we get a non-trivial automorphism after applying Spec which we'll call the same thing $\sigma: S\to S$. Thus $Aut(S)$ contains at least the enormous group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ (it is in fact equal to this).
This is hardly what we would like to call morphisms coming from geometry. This is what happens when we just work in terms of (non-relative) schemes. We seem to pick up all sorts of morphisms coming from number theoretic or arithmetic sources. Now let's consider $S$ as a scheme over $S$ with the trivial strucutre map $id_S: S\to S$.
Now if we check whether or not any non-trivial $\sigma: S\to S$ is in the automorphism group of $S$ as an $S$-scheme we see it can't be because the appropriate diagram will not commute. In the category of $S$-schemes we see that we actually kill off all the non-geometric automorphisms and $Aut_S(S)$ is just the single identity map we thought we should get.
At first, it looks like considering this more complicated category will make things ... more complicated, but in practice we usual do exactly the situation above. If $X, Y$ are varieties over $k=\overline{k}$, then we work in the category of $k$-schemes. It will simplify things by killing off the non-geometric morphisms.