Why is the method of separation of variables is the only method used in physics for solving partial differential equations despite that they are not the most general solutions? Do these solutions form some kind of complete set? Is there a proof?
Fourier's separation of variables technique was one of the earliest methods for solving linear partial differential equations. A good number of classical problems such as the Laplace equation, the Heat equation, and the wave equation can be solved this way, and the solutions are explicit. (By the way, $x+y = x\cdot 1 + 1\cdot y$ is a separated function.) There are not a lot of other techniques that can be used to find explicit solutions; so the technique remains useful for gaining theoretical and practical information about these classical equations of Engineering and Physics.
It is also, in connection with PDFs (Probability Density Functions) a reason of independence for example in the simplest case where the variables are $x$ (space) and $t$ (time) when we know that there is no connection between space and time.
Indeed a PDF $f(x,t)$ can be written $f_1(x)f_2(t)$ if and only if the associated Random Variables $X$ and $T$ are independent. And this is frequently the case.