Why not use all the colours in colorimetry?
Why not send in white light (with all the wavelengths at once)?


This is certainly and routinely done today. The main thing is how much price are we willing to pay? In a rigorous sense, colorimetry refers to the fact that we use optical filters to isolate wavelengths rather than monochromators. When the latter are used, you call them spectrophotometers. Basically, the question is referring to scanning instruments versus diode array or multiplexing instruments. Both exist today and both have their advantages and disadvantages.



In scanning spectrophotometers, one wavelength passes through the sample at a time. In diode array instruments, all light passes through the sample at once and it is dispersed later. The dispersed light falls on an array of detectors (thousands of tiny detectors, usually 1024). The detectors do not know what wavelength is falling on them, however, they are calibrated. 

Similarly, if you were working in the 1970s-80s you would use a scanning infrared spectrophotometer (one wavelength at a time). One needed a prism made of salt or a very special diffraction grating to disperse the infrared radiation. This made life very difficult. Imagine working with a salt prism!

In Fourier transform infrared spectroscopy, all wavelengths are passed at once, but there is no array of detectors. Unfortunately, there is no detector which can differentiate wavelengths on the basis of their frequency and separate them. We need to modulate them in such a way that an ordinary detector can "follow" the frequency of wavelengths. Here comes in the Michelson interferometer. The raw output does not look like a spectrum at all, but it has all the information "hidden" there. You need Fourier transform to recover a typical infrared spectrum.

Now one may ask why don't we use FT for UV-Vis? The answer is that the required precision of a moving mirror is so high that it is rather difficult to make one.