Why is doing fft on a non-stationary signal a problem?
The frequency interpretation on a non-stationary signal is not "incorrect".  For example, it still works for doing convolution/filtering.  However, it becomes opaque with regard to reflecting features interesting for analysis.

As a simple example, a morse signal of fixed carrying frequency is basically signal times carrier in the time domain and since the carrier is represented by two pulses in the frequency domain, the total frequency representation is an "echoed" variant of the transform of the signal.

However, the signal is interesting as a sequence in time.  Its Fourier transform is an obfuscation rather than a recovery of information.

For many non-stationary signals, the information you are looking for is contained in the sequence of parameter changes over time: an overall Fourier transformation hides those further under the surface than they were before.

There are "number theoretic transforms" also obeying the convolution theorem of the Fourier transform and based on powers of unit roots.  They have FFT-like algorithms and data flow and can be used for convolving number sequences.  They are utterly useless for any form of analysis since their "transforms" invariably look like pseudo-random noise.

In this case, we are talking about something which is "correct" and thus applicable in given situations but which is not useful as an analysis tool.  To a lesser degree, that is the situation when using a Fourier transform encompassing all of a non-stationary signal.