Meaning of type inference rule for abstraction in lambda-calculus
The problem is that whatever the context $\Gamma$ may imply about the
variable $x$ is irrelevant. This rule is intended to type a
$\lambda$-abstraction, and the variable $x$ may be changed to another
variable by $\alpha$-conversion, without changing the semantics of the
$\lambda$-abstraction,  and thus without changing its type.

By writing $\Gamma, x: \tau_1$, we are considering a context $\Gamma$
where the variable $x$ has type $\tau_1$, ignoring what the type of
$x$ may have been in context $\Gamma$ alone. If $x$ could be assigned
a type in the context $\Gamma$ alone, it would refer to some homonymous variable that is irrelevant to the abstraction, since its scope is shadowed in the abstraction by the local variable $x$. Indeed, an $\alpha$-conversion would make $x$ disappear from $e$ altogether.

In other words: $\Gamma \vdash x: \tau_1$ may or may not be true, but
it is simply irrelevant to typing the abstraction.

Then, it is possible that $\Gamma \vdash e: \tau_2$. But that is only
possible if the type of $x$ is irrelevant, for example when $x$ does
not actually occurs in $e$, as for the abstarction $\lambda x.42$,
Then $e=42$ which is indeed typable. But in general, you cannot type $e$ without having a type for $x$ (for the variable $x$ of the abstraction).

In other words, you expect the type of $e$ to be in general related to the type of $x$, and it is their relation that the rule is intended to express.

What the rule says is that, if by adding to the context $\Gamma$ that the new
variable $x$ has type $\tau_1$, you can infer then that $e$ has type
$\tau_2$ in this new environment, then the abstraction $\lambda x. e$
has type $\tau_1 \rightarrow \tau_2$ in the original environment
$\Gamma$.

It says that the type of $e$ is dependent on the type $x$ may have.

Then, wherever the abstraction is applied to another expression $e'$, the type of the application is whathever the type of $e$ should be when the type of $x$ is that of $e'$.  But that is another typing rule.