To answer this question, we need a strict definition of what is classical music in a broader sense. I am not sure such a definition exists, or could ever exist. Many composers are somewhere in between. The "King of Ragtime" is one of those cases, flirting with classical idioms but not necessarily committed to the classical tradition.I guess a critical question to ask is what he has mostly composed and what he is known for. If he has written like 2 symphonies 1 opera but 300 rags, well lets put it this way: he is not the archetype of a classical composer. Some people say Gershwin is jazz, which is wrong, as jazz is basically improvisation. His compositions are clear, note for note (if people improvised on them later, is a different story). Gershwin is using elements of jazz, but his classical training can't be denied. So another case which is difficult to categorize but for different reasons. Skin color surely played a role in Joplin's publicity and recognition at that time, but I doubt this is the reason he is not generally considered classical by today's standards.
Part of the problem is that musical genres can't be compared qualitatively. Saying for example: "Rachmaninoff is better than Eminem" is nonsensical, because you have to say in what, and the what is so different that it can't be compared. In this case, we have a genre called classical and like you mention, there are some attributes which resemble the classical style. But classical does not say anything about the quality of the music. There are a lot of classical composers who could not come up with anything special or who composed, simply put, trash. So why aspire to be called classical? To be called "king of genre X" on the other hand, does say something about quality, and that's how Joplin went down in history.
The reason ragtime is not taught in conservatories I guess is because the form is simple,it is technically not very challenging and has not so much musical diversity/variety of emotions as classical. It was music designed to entertain. What he would write if the circumstances were different, is speculation of course. But he proved to be very inventive and successful within the confines of ragtime, so imho, he would be brilliant in classical forms as well.
However SJ's best work (the rags) fit in the ABAtrio style drawn from circus marches, which were popular in his time. It is a valid format, extremely popular in the late 18th & early 19th century, but with the masses, not the upper class. Thus the "highbrow" and "lowbrow" distinction.