You seem to be looking for a single unifying idea that explains why $L$-functions are the way they are. I don't think this is possible. $L$-functions are more like an elephant, with different perspectives offering different insights.
For example one can begin with the $L$-functions of Galois representations, motivating them as a way to prove analytic properties of the Galois representations (e.g. Dirichlet's theorem, Sato-Tate conjecture). This requires holomorphicity or meromorphicity of $L$-functions, which motivates the search for an alernative expression for them (e.g. as an integral). Automorphic $L$-functions then appear as another class of integrals which admit Euler products and thus can potentially match Galois $L$-functions, and we look for the most general definition of automorphic $L$-functions to have the best hope of matching every $L$-function on the Galois side.
On the other hand, one can take a perspective from representation theory, where the fundamental problem is classifying automorphic representations. Automorphic $L$-functions are then a natural invariant of automorphic representations that could be useful in the classification. Galois representations are seen as parameterizing automorphic representations and the $L$-function is needed to see which matches to which.
One could also start in the function field context where $L$-functions calculate the eigenvalues of Frobenius on the cohomology of a sheaf on a curve. Then $L$-functions of Galois representations could be understood as an attempt to extend this to a context where cohomology cannot be defined and $L$-functions of automorphic forms as an even further extension.
In many ways, some of the major questions in the Langlands program concerns constructing correspondences between these different structures in such a way that they preserves certain properties and their corresponding $L$-functions match up.
Note that when matching $L$-functions we are really concerned with matching their local Euler factors rather than any global object. At unramified places we are matching Frobenius eigenvalues to Hecke eigenvalues via the Satake isomorphism. This can be justified purely locally (e.g. via the geometric Satake isomorphism) with no reference to $L$-functions.
My question simply is whether $L$-functions are more than just a signature (which could possibly easily be replaced with another signature in the future)?
This is already two questions. I think what you mean by the first one is whether the $L$-function is more than some invariant used to match Galois representations, automorphic forms, and motives in the Langlands correspondence.
The answer is yes, of course, $L$-functions are more than this. For example $L$-functions are used to prove analytic properties of Galois representations or automorphic forms (like equidistribution). Their special values give useful formulas for other invariants (like the class number formula or the BSD formula).
Could they be replaced as a tool to match Galois representations and automorphic forms? Absolutely. For example Vincent Lafforgue's Langlands parameterization of automorphic forms over function fields doesn't directly use $L$-functions, though the excursion operators involved are often related to $L$-functions. The Fargues-Scholze approach to local Langlands doesn't really use $L$-functions. One can state the global Langlands conjecture roughly as the existence of a matching between global automorphic forms and Galois representations compatible with this local Langlands correspondence, and that doesn't use $L$-functions.
On the other hand, could $L$-functions be replaced in everything they do? Surely not.
Does it somehow encode deep rooted properties of these different structures?
Of course. See the before-mentioned BSD and class number formulas, as well as their generalizations, which describe how some deep-rooted properties are encoded in $L$-functions. On the automorphic side, $L$-functions can give the constant term of Eisenstein series, so zeroes of $L$-functions determine when Eisenstein series limit to cusp forms. This is especially important for $p$-adic $L$-functions which describe $p$-adic congruences between Eisenstein series and cups forms - these help in various ways to study Galois representations and automorphic forms.
And, what motivates someone to define the $L$-function of these structures in a certain way - are we just defining it the way we do, because we know that that is the best chance we have to get these different correspondences we care about?
The definition of $L$-functions are partially motivated by the correspondences but that isn't the only motivation. Dirichlet $L$-functions were originally motivated by analytic reasons. The correspondence in this case is the Kronecker-Weber theorem, which $L$-functions aren't really helpful to state or prove. On the other hand Artin defined his $L$-functions with the goal of finding a non-abelian class field theory, and thus was arguably motivated by the goal of finding a correspondence although he didn't know what kind of correspondence he was looking for. After these constructions the motivations for many different definitions of $L$-functions was probably largely by analogy - "it worked in this case so let's try it in this other case and see if we get anything useful".
