The full question probes as far as "what is science?", so I'll try simplifying.
Fourier Transform is a tool. A mathematical construct. The goal is to accurately describe reality.
Suppose a swinging pendulum. Suppose we know it swings 3 times per sec because we designed a motor to drive it such. How do we describe this swinging mathematically? We can say,
$$
s(t) = \cos(2\pi \cdot 3\cdot t) \tag{1}
$$
and this would accurately describe the swinging over any arbitrary duration $t_0$ to $t_1$. The 'mathematical construct' here is a continuous-time function that does not assume a predefined domain but rather permits it ($t_0$ to $t_1$) to be selected on demand.
Now suppose despite knowing the swing rate, we seek to measure it and describe it mathematically from those measurements. We record pendulum positions for 5 seconds, call it "data A". Then for 12, call it "data B". We take the Fourier Transform of A and B, and get different results despite the physical process being exactly the same; how so?
Because by taking the Fourier Transform of $x$ we're answering the question, "what is the continuum of frequencies of continuous complex sinusoids spanning all time, $-\infty$ to $\infty$, that would infinitesimally sum (integrate) to $x$?" This continuum changes simply because in B we sum to non-zero over a greater interval than in A. Now let me ask this: why do you care?
Must we really measure a process from big bang to heat death to know its frequency? No. How much must we measure? That's a question of statistics, and statistical significance. Given a 12 second measurement, applying our knowledge of reality (physics), we can state confidently: "this pendulum will swing at a rate of 3 cycles per second given same environment (wind, gravity, etc) and driver (motor)". Note this statement will not change even if we did measure for $10^{100}$ years; it'll just say "we got a poor bloke to stare at a pendulum for all eternity doing exactly what we knew it'd do."
Now suppose we don't know the frequency ahead of time; the drill's the same: measure for "long enough", generalize. For pendulum, we measure for 12 secs - if there's wind, maybe we measure for 60. Then apply knowledge of the system, such as (in noiseless case) "pendulum swings at only one frequency", to clear "artifacts" like spectral leakage - and we can again arrive at a formulation like $(1)$.
Same for audio; for speech we've studied vocal chords, etc physically and statistically to know "long enough" - i.e., measure until it repeats (one full period).
Why favor DFT?
Because unlike continuous FT, the Discrete Fourier Transform does not assume infinite duration basis functions. You've measured for 12 secs? The basis functions are 12 secs. No need to try to describe what happens beyond those 12 secs, nor assume. The results are simply meaningful, and don't violate assumptions when we decide to generalize to a $(1)$-like formulation.
Is "no physical signal bandlimited"?
Depends what "bandlimited" means. Bandlimited = finite range of frequencies. The question then is, what's a frequency? or frequency of what? of infinite duration sinusoids? Then yes, no physical signal is bandlimited. But then, again, why do you care?
Asserting this is plain misleading as it suggests every physical process has infinite derivative processes each at their own frequency. Knowing the actual max frequency of a process is a physics endeavor, not transforms'. The DFT does not commit this fallacy (but its weakness is blindness to anything beyond half its sampling rate (which is a non-issue if we know said frequencies don't exist)).
Why not favor DFT or CFT?
Because the building blocks are inherently limited in kinds of behaviors they can describe 'directly'. Suppose the same pendulum, but now damped. Its FT:
$$
s(t) = e^{-t} \cos (25t) u(t)\ \Leftrightarrow\ S(\omega) = \frac{1 + j\omega}{(1 + j\omega)^2 + 625} \tag{2}
$$
What does the Fourier Transform tell us? Infinitely many frequencies. Is this a sensible physical description? Hardly (only in certain indirect senses); the problem is, FT uses fixed-amplitude sinusoids as building blocks, whereas here we have a variable amplitude that cannot be easily represented by constant frequencies, so FT is forced to "compensate" with all these additional "frequencies".
We require non-stationary methods that can map out frequency and amplitude that changes over time. Below is the synchrosqueezed continuous wavelet transform of $(2)$:
Clarification: DFT vs CFT
DFT doesn't exactly escape the flaws of CFT in computed values; indeed DFT is a sampling of DTFT, and DTFT is a periodization of CFT - that is, we can predict DFT values from CFT. DFT can hide inconvenient frequencies (e.g. by sampling perfectly integer periods of a waveform), but they're exposed by trimming (or padding) the input.
The advantage stems from interpretation of resulting coefficients, whose basis functions aren't by definition infinite-periodic. We interpret them as strengths of correlations with bases of input's length (that, for example, enables generalizing the 'convenient' result of a single nonzero DFT bin as a perfect infinite sinusoid).
Black box case
If we know nothing about the system, then to map its frequency with certainty we must not only sample for all time but at an infinite rate. We can do neither. At bare minimum we begin by making assumptions, then refining with further observation - the empirical way. Refer to MBaz's answer + comments for elaboration.
Morale
Fit abstract constructs to reality, not vice versa.
