There is a lot to say here, so I will break my answer down in stages:
$(1)$. Is it possible to construct solutions to a PDE by making a sequence of a change of coordinates and then eventually write the solution in a Fourier series?
Answer: N0
This technique might work for some linear PDEs, but this would not work for non-linear PDEs, even ones as simple as the non-linear transport equation (Burger's equation) $u_t+uu_x=0$.
$(2)$. Is it possible to solve PDEs without using the tools from real analysis/set theory?
Answer: "No", and the exceptions are not as useful
Analysis, especially Functional Analysis is extremely important to PDEs. When you want to calculate the "energy" of a solution, you need to understand how a function's energy is related to other quantities. Typically, this is based on the concept of the norm. Without analysis, we don't really have any good way of analyzing "energy" of a solution. In addition, we could not comment on regularity/smoothness. If we get a solution to a PDE, does it have a derivative, jump discontinuities? These questions are able to be answered in some cases, but not in general, and all use analysis.
In addition, when studying an unknown PDE, you may not even know a solution exists. For some PDEs, there might be some "tricks" to transform the PDE to a simpler one, but for a general non-linear PDE, you may be in the dark. Before we actually try to construct a solution, sometimes we just want to show one exists. This often requires using "fixed-point" methods.
What this does is takes an operator $T$ on some function space, and maps it to another function space. If the mapping is a contraction, then we have a fixed point, and a solution is guaranteed to exists.
Mathematically, suppose we wanted to find a a solution to the Heat Equation on an infinite metal bar. Such a function must have at minimum two continuous derivatives (solutions to the Heat Equation smooth out, but this is another topic). Functions which are continuous everywhere on the real line, and continuous for a time greater than $0$ can be all grouped together into one set:
$X_1=C^2(\mathbb{R} \times [0,T])$. All functions which obey the property exists in this space, so any possible solution to the heat equation would lie in this space. Another question you can ask is, does the solution lie in a more general space, such as one that includes $X_1$ as a subset? This has a well developed theory as well.
So to prove a solution to the heat equation exists, you want to construct a contraction mapping on $X_1$ using some suitable operator. Again, the details require norms, which is all analysis. If the operator $T:X_1\rightarrow X_1$ is contractive, then you have a fixed point, which means a solution to the PDE exists (even if we cant construct it explicitly).
Next, PDEs study functions with derivatives, which is literally a concept in analysis. You might sometimes be able to solve PDEs with just advanced calculus techniques, but these techniques do not address the idea of uniqueness of solution, regularity of solution and other important questions. For example $u(x,t)=x^2t+2t$ is a polynomial which solves the heat equation in the sense that differentiating $u(x,t)$ will solve the heat equation, but it doesn't solve it an interesting way. This relates to the topic of well-posedness. When studying PDEs, you dont want an infinite number of solutions, which can be constructed by multiplying the polynomial I gave via some suitable constants.
We also want the solution $u(x,t)\rightarrow 0$ as $t\rightarrow \infty$. This means the solution will "vanish at infinity", so the value of the function decreases at time goes on. Functions that "vanish at infinity" have their own space as well, so yes, more Analysis to study them. Polynomials do not vanish at infinity.
Finally, sometimes it is very challenging to solve certain boundary conditions. This is especially true in higher dimensions. Here is an example of a non-linear wave equation in three dimensions for illustration, but the example I present can be made far more complicated in higher dimensions.
There is no standard way to do a coordinate transformation that will make a complicated PDE easier to solve.
$\begin{cases} u_{tt}+(\lambda u^2)u_{xx}=0 \\ u(0,0,t)=\phi(x), u(1,0,t)=\psi(x) \\u(x,y,0)=h(x) \end{cases}$
Let $C(x)$ denote the Cantor function.
Define $\phi(x)$ as $\phi(x)=C(x), |x|\leq1$ and $\phi(x)=0$ for $|x|>1$, and $\psi(x)=sin(exp(c|x|^2))$, $|x|\leq 1$ $\psi(x)=0, |x|>1$
It is very easy to cook up PDEs which are catastrophically difficult to solve even numerically, let alone analytically. Coordinate transformations wouldn't likely be of much use if your boundary conditions involve odd functions such as the one I constructed
Sometimes you use calculus-themed techniques and clever transforms, but no, most of PDEs requires some use of analysis, and certainly all general properties of PDEs are understood through the context of analysis.
Also note coordinate transformations often involve differential geometry techniques, which are equally as sophisticated in their own right.
