Your question is partly about History. And the History of how Mathematicians were led to consider orthogonal expansions in trigonometric functions is not a natural one. In fact, Fourier's conjecture that arbitrary mechanical functions could be expanded in a trigonometric series was not believed by other Mathematicians at the time; the controversy concerning this issue led to banning Fourier's original work from publication for more than a decade.

The idea behind trigonometric expansions grew out of looking at the wave equation for the displacements of a string:
$$
\frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2 u}{\partial x^2}.
$$
In 1715, B. Taylor, concluded that for any integer $n\ge 1$, the function
$$
u_n(x,t)=\sin\frac{n\pi x}{a}\cos\frac{n\pi c}{a}
(t-\beta_n)
$$
represented a standing wave solution. $n=1$ corresponded to the "fundamental" tone, and for $n=2,3,\cdots$, the other solutions were its harmonics (which is where the term Harmonic Analysis first arose in Mathematics.) It was a natural question at the time to ask if a general solution could constructed from a combination of the fundamental mode and the harmonics. If such a general solution were to exist in the form
$$
u(x,t) = \sum_{n=1}^{\infty}A_n u_n(x,t),
$$
where $A_n$ are constants, then it would be necessary to be able to expand the initial displacement function as
$$
u(x,0) = \frac{a_0}{2}+a_1\cos\frac{\pi x}{a}+b_1\sin\frac{\pi x}{a}+\cdots.
$$
The consensus at the time was that an arbitrary initial mechanical function could not be expanded in this way because the function on the right would be analytic, while $u(x,0)$ might not be. (This reasoning was not correct, but Mathematics was not very rigorous during that era.) The orthogonality relations used to isolate the coefficients were not discovered for some time after that by Clairaut and Euler.

Fourier decided that such an expansions could be done, and he set out to prove it. Fourier's work was banned from publication for over a decade, which tells us that the idea of expanding in a Fourier series was *not* a natural one.

Fourier did not come up with the Fourier series, and he did not discover the orthogonality conditions which allowed him to isolate the coefficients in such an expansion. He did, however, come up with the Dirichlet integral for the truncated series, and he did essentially give the Dirichlet integral proof for the convergence of the Fourier Series, though it was falsely credited to Dirichlet. Fourier's work on this expansion became a central focus in Mathematics. And trying to study the convergence of the trigonometric series forced Mathematics to become rigorous.

What Fourier did that was original is to abstract the discrete Fourier series to the Fourier transform and its inverse by arguing that the Fourier transform was the limit of the Fourier series as the period of the fundamental mode tended to infinity. He used this to solve the heat equation on infinite and semi-infinite intervals. Fourier's argument to do this was flawed, but his result was correct. He derived the Fourier cosine transform and its inverse, as well as the sine transform and its inverse, with the correct normalization constants:
\begin{align}
f & \sim \frac{2}{\pi}\int_{0}^{\infty}\sin(st)\left(\int_{0}^{\infty}\sin(st')f(t')dt'\right)ds \\
f & \sim \frac{2}{\pi}\int_{0}^{\infty}\cos(st)\left(\int_{0}^{\infty}\cos(st')f(t')dt'\right)ds.
\end{align}
He used these to solve PDEs on semi-infinite domains. The sin's and cos's were eigenfunctions of $\frac{d^2}{dx^2}$ that were obtained using Fourier's separation of variables technique. The term "eigenvalue" grew out of this technique as a way of understanding Fourier's separation parameters.

Based on this story, I would say that it was not a natural idea to expand a function in trigonometric functions. Fourier's work led to notions of linear operators, eigenalues, selfadjoint, symmetric, and general orthogonal expansions in eigenfunctions of a differential operator, but it took over a century for this work to look "natural."