What is reactivity really, and can it be quantified?
I think the term "reactivity" is often used to denote kinetic lability, a proclivity to react.  Let's look at the two reaction coordinates shown below.  They both represent the interconversion of molecules A and B through intermediate I.



Note that the potential well that I exists in is much deeper in the drawing on the left than the drawing on the right.  Therefor the activation energy for I to escape will be much larger for the intermediate on the left than the intermediate on the right.  The intermediate on the right would be said to be more reactive (have a shorter lifetime) than the intermediate on the left, because it requires less energy for it to escape.

An experimentalist can infer the depth of the well through experiments.  Through such experiments the experimentalist learns that free radicals, highly strained hydrocarbons, carbenes, etc. exist in potential wells that must be very shallow because they react so readily.  Using this empirical data the experimentalist can then infer how similar compounds are likely to react. A computational chemist can infer the same thing by calculating the profile of the reaction coordinate for the various reactions under investigation by the experimentalist, or as is becoming increasingly common the computational chemist can predict what should be and let the experimentalist confirm those predictions.