You’re absolutely right, although in my experience 3 C is not a huge span when it comes to m.p. determinations. Even a 1% impurity can throw off a measurement, and that’s why melting points are often given as a range rather than a single number in the literature.

Nowadays, you absolutely can run an unknown sample against a test battery and find composition and quantity down to a pretty good degree. There are test kits for the powder/solution as well as for blood samples. In the case of blood samples they also focus on metabolites. The motivation is that if someone arrives at the hospital with an adverse drug reaction, if they have used illicit drugs the only way to figure out what they have in their system may be to check against the metabolites in their blood.

The gist of it is that the chromatography equipment will separate the components into “pure” fractions, which are called “peaks”. That will be a Liquid Chromatograph (LC), its High Pressure version (HPLC), or for some sample matrices a Gas Chromatograph (GC).

What actually happens is that different components move at different speeds throughout the column, so they leave the column (“elute”) at different times. Commonly this stream is then connected directly to a detector, and for this type of application the Mass Spectrometer (MS) is the workhorse.

It works by turning the sample into a vapor and then bombarding it with electrons which causes the molecules to disintegrate in a signature pattern. The stream of ionized molecule fragments is accelerated perpendicular to a voltage (like how an old CRT TV works), which means that fragments with lower mass will change their trajectory more than fragments with higher mass. (The charge of each fragment also matters here.)

Finally the fragments hit a detector array which registers counts for the different fragment masses that were in the molecule for that fraction from the chromatograph.

Here’s the cool part: the mass fragment signature is such a good fingerprint of a molecule that it can be compared to a database with tens of thousands of known substances, and usually give a pretty good match. Also, an analytical chemist that is experienced in MS can often figure out what molecule they’re looking at just from the mass spectrum and maybe a couple of simple hints.