Why are Fourier analysis and transform only applicable for LTI systems?
The family of Fourier Transforms are specificaly developed for analysing frequency contents of the signals for which there is no definition of linearity or time invariance. Hence we can define the Fourier transform of any signal, as long as it's integrable (i.e. stable).

On the other hand we can also define a Frequency Response $H(\omega)$ for a particular class of systems (those being LTI) when they are characterised by a unique signal denoted by $h(t)$ which is called as the impulse response of the LTI system. 

The Fourier transform $H(\omega)$ of the impulse response $h(t)$ is important because the complex exponentials $e^{j\omega t}$ are eigenfunctions of such LTI systems and when the input is decomposed into a sum of such complex exponentials, via a Fourier transform, so will be the output, multiplied with a complex gain given by the Frequency Response $H(\omega)$ applied for each particular frequency. This gives a considerable ease for the development of the mathematical theory of signals and systems.

On the other hand consider the following nonlinear system defined as:
$$y(t) = x(t)^2$$
For which you can apply CTFT to both sides to derive the result in frequency:
$$Y(\omega) = \frac{1}{2\pi} X(\omega) \star X(\omega)$$

Also consider the following linear but Time-Varying system defined by:
$$y(t) = g(t)x(t)$$
For which you can also express the relation in Frequency domain via CTFT as:
$$Y(\omega) = \frac{1}{2\pi} G(\omega) \star X(\omega)$$

Which I hope convinces you about the scope of Fourier Transforms and their applications to systems other than LTI.