Sfrgttk

Adjectives derived from proper nouns are known as proper adjectives, and are capitalized:

A piece of writing could be Shakespearean, not shakespearean.
A person may be Canadian, not canadian.

Even Chrome's spellchecker sees these as correct and incorrect.

However, quixotic is written in lower case, despite coming from the name of the character Don Quixote. Similarly, draconian laws are named for Draco, a particularly brutal senator from ancient Athens.

Does anyone know the reasoning behind some proper adjectives not being capitalized in common usage?

As you correctly say, technically words associated with a proper noun should be capitalized.

However as time and usage goes on, these words tend to become words in their own right, not associated any more with the person they are named after.

So 'Shakespearean' means 'associated with or like Shakespeare'. It has no meaning apart from the association with the person.

On the other hand 'quixotic' is defined as "foolishly impractical especially in the pursuit of ideals. especially : marked by rash lofty romantic ideas or extravagantly chivalrous action". The definition makes no reference to Quixote, and people can (and do) use the word without knowing who Don Quixote is. This is even more true with 'draconian', which I had no idea was related to a person. Over time the adjective morphs from directly referring to the person etc, to indicating characteristics that were once associated with that person but are now considered independently.

So the answer is that the word loses its initial cap when it stops being associated with the person (or thing) it was originally associated with.

In a comment posted years ago to the question Why is "biblical" the only proper adjective to not use upper case? I listed some other exceptions to the general rule that the first letter of an adjective derived from a proper name is normally capitalized. Only the letters q, w, and x faild to yield an example:

arabesque, byzantine, caesarean, draconian, epicurean, faradic, galvanic, herculean, italic, jesuitical, kabbalistic, lilliputian, mercurial, nazi, oedipal, pyrrhic, rubenesque, spartan, terpsichorean, utopian, voltaic, and zephyrous

Why do these exceptions occur? The not-very-satisfactory answer seems to be that common usage determines whether an adjective based on a proper name is initial-capped or lowercased. Dictionaries provide their spelling preferences based on what amount to the found objects of preponderant real-world usage in each case; and thenceforth, real-world usage (to the extent that it is influenced by people who look things up in dictionaries) tends to reflect the dictionary treatment. The very circularity of the process makes it extremely difficult to determine where and when the critical decision regarding initial cap versus all lowercase got made.

I don't see how else to explain why Oedipal is commonly initial-capped when it refers to the mythical character Oedipus (as in "Oedipal resistance to fate") but commonly lowercased when it refers to Freud's Oedipus complex (as in "oedipal feelings"), even though the complex is explicitly named after Oedipus.

Bryan Garner, Garner's Modern American Usage, second edition (2003) concludes that trying to make sense of exceptions to the normal rules about what gets capitalized and what doesn't is a fool's game:

There is simply no way to reason out why Stone Age is capitalized but space age is usually not, why October is capitalized but autumn is not, why in scientific names the genus is capitalized but the species is not—even when the species name is derived from a proper name {Rhinolophus philippinensis}.

Ultimately, capitalization conventions rest on strong general tendencies tempered by exceptions that are neither consistent nor explicable.