What is the purpose of UV-VIS Spectroscopy?
The broad absorption bands of the electronic transitions observed in UV/Vis spectroscopy/spectrophotometry, as well as the myriad types of electronic transitions that might lead to absorption at a given wavelength, indeed make the technique much less powerful for identification of unknown compounds as compared to, e.g., IR, NMR, microwave, and mass$^*$ spectroscopies.

Jon Custer is correct in his comment to the question that UV/Vis spectra can provide key information in identifying unknown species.  However, I know of only two approaches to obtaining predictions of a given UV/Vis spectrum:


Experimental spectra of compounds suspected to be structurally related to the unknown species
Quantum chemical computation of the electronic transition spectra of candidate species


The former approach requires the relevant experimental data to exist; the latter is not trivial to perform accurately. Both suffer from significant uncertainties: How similar is a reference experimental spectrum to that of the unknown species? Can I trust that my quantum computation will give realistic band locations and strengths?



In my experience, the strength of UV/Vis lies in its facility for inexpensive, convenient, routine, non-destructive quantitation of known solutes, with known absorption spectra and/or extinction coefficient(s) at given wavelength(s).  The following list of strengths is almost certainly incomplete:


The instrumentation and consumables are relatively inexpensive.
No auxiliary cooling, gas supply, etc. are required.
Preparation and analysis of fixed-volume samples is trivial: pour liquid into cuvette; place cuvette into instrument; measure.
Continuous analysis of dynamic systems (e.g., chromatographic eluates and ongoing reactions) is possible using a flow cell.


A simple example is described here, unfortunately paywalled.
A wide variety of configurations is presented at a commercial vendor here.
(No endorsement intended. There are numerous other suppliers out there; I chose this one because it had a good variety shown in schematic.)

Interference from most solvents is minimal, particularly in the visible range.
Beer's Law in its linear form has a relatively high dynamic range for most UV/Vis-active solutes$-$that is, the concentration where absorbance deviates from linearity is much greater than the minimum concentration detectable:


$$
\frac{C_{\max}}{C_{\min}} \gg 1
$$


Another implication of Beer's Law is that interference(s) from a small number of absorbing co-solutes are relatively straightforward to handle, either by measuring at isobestic points or by measuring multiple wavelengths and solving the linear absorbance equations for the unknown concentrations. For example, if two species $\ce{X}$ and $\ce{Y}$ are present and absorbances are measured at wavelengths $\lambda_1$ and $\lambda_2$, the concentrations can be found by solving the following two expressions:


$$
A_{\lambda_1} = \epsilon_{\ce{X},\lambda_1}\cdot C_\ce{X} + \epsilon_{\ce{Y},\lambda_1}\cdot C_\ce{Y}
$$
$$
A_{\lambda_2} = \epsilon_{\ce{X},\lambda_2}\cdot C_\ce{X} + \epsilon_{\ce{Y},\lambda_2}\cdot C_\ce{Y}
$$


The steady-state nature of the measurements allows for 'deep' examination of a particular sample.


The full UV/Vis spectrum for a given volume of sample can be measured, giving a rich "fingerprint" of the particular solution and the solutes in it
The absorbance at a particular wavelength can be tracked over time, allowing for detailed kinetic analyses

At least in the visible range, the quality of the substance being measured is directly perceptible to the human senses. This is actually quite viscerally satisfying, to see that a sample anticipated to contain significantly more of a solute does indeed produce a noticeably stronger coloration of the solution.


While IR and microwave spectroscopies share some of the above advantages, they are far less suited for analysis of solutes, as the solvent absorbance is often much more likely to confound measurements.  Further, the instrumentation and sample preparation are generally much more involved.

Mass spec is terrifically expensive and complicated, is an intrinsically destructive method, and is less well suited for direct quantitation since there is no way to know a priori what fraction of the injected sample is actually making it to the detector and thus careful calibration is required.

NMR, though non-destructive, also requires quite expensive equipment that is complicated to operate properly, and interpretation of the spectra requires quite a bit of practice.

I know too little about EPR/ESR, EXAFS, XANES, and others to comment on them here. I believe the above strengths hold in comparison to those methods as well, however.



$^*$ Technically, MS is mass spectrometry, not spectroscopy, since the measurement is of discrete ion impacts on a detector, as opposed to absorbance of portions of the electromagnetic spectrum by a detector.