One thing of which I am thoroughly convinced is that there is no such thing as a perfect philosophy. By that I mean a philosophy free from any knotty problems or apparent contradictions. If someone claims to have such a philosophy, what it means, invariably, is that the philosophy has not been thought through enough to make the problems apparent. Chesterton's dictum that "nine out of ten new ideas are old mistakes" is appropriate here.
I believe one reason people avoid classical philosophy is because, having been thoroughly thought through, its problems have been exposed and are apparent to the uninitiated. Plato's philosophy, for example, struggles with the problems of the ontological status of the Ideas (where exactly do these things exist?) and their relationship to the physical world (how do physical beings "participate" in Ideas?) The long history of struggling with such questions led to Aristotle's revision of Platonic philosophy, through the philosophy of the Middle Ages, and culminated in the synthesis of St. Thomas Aquinas, each step having its own peculiar difficulties. Since the opponents of classical philosophy are quick to home in and advertise the problem areas, even those only passingly acquainted with classical philosophy are likely to know what they are.
But another way to look at the situation is to recognize that there won't be any surprises in classical philosophy. Whatever the difficulties are (and there will always be difficulties), they will have been smoked out by the centuries of philosophical reflection. There is virtually no chance that a philosopher will come along who notices some grave difficulty that hadn't already been noticed; anyone who thinks he has stumbled across such a thing only proves that he is not familiar with the history of philosophy. In vulgar marketing terms, in classical philosophy you can rest assured you will get a product that has been thoroughly tried and tested. (And not just in abstract philosophical reflection; classical philosophy has been tested in the court of life as well, forming the foundation of Western civilization in Greece and sustaining it for millennia through the Middle Ages).
It is tempting, in the face of some of the difficult problems classical philosophy has struggled with, to abandon the tradition altogether and start afresh from a clean state. This is essentially the attitude that gave birth to modern philosophy, most explicitly stated in Descartes. The experience can be heady, but it soon becomes apparent that the modern philosopher has only exchanged one set of problems for another. Descartes may have been satisfied with his philosophy, but the philosophers who followed him certainly weren't. Yet rather than drawing the lesson that it may have been foolish to abandon the classical tradition in the first place, modern philosophers adopted an attitude of permanent revolution. Each one starts philosophy afresh, convinced that his effort, finally, will put philosophy on the one absolutely sure footing. The most serious effort in this regard was that of Immanuel Kant, who was convinced he had established with his "critical philosophy", once and for all, the permanently sure foundation of philosophy. Alas, his followers were not convinced, but only drew the lesson that they themselves must begin philosophy yet again anew. The history of modern philosophy is a form of degenerate tradition; not a tradition that absorbs and organically grows an ongoing project of philosophical knowledge, but a "tradition" that repeatedly rejects as problematic all that came before and starts philosophy afresh. The hope of a modern philosopher is not to understand his philosophical predecessors and expand on their reflections, but to discover a "revolutionary and new" technique or principle in terms of which philosophy must be recast, and which will free philosophy from the problems discovered in the last revolutionary cycle. Since the problems latent in this new technique are as yet undetected, this ignorance offers the philosopher the illusion of the hope that he has finally "solved" philosophy. This continual cycle of creation, revolution and destruction is what gives philosophy its bad name in the modern world; it actually does go nowhere as its critics claim.
In any event, the upshot is that there isn't much point in trying to convince someone of the value of classical philosophy who is under the illusion that he possesses a problem-free modern philosophy. Classical philosophy is preferable to modern philosophy because the problem areas of classical philosophy are the problem areas of reality; trying to escape them is as futile as trying to escape from reality. But the appropriate response to someone attempting to escape from reality is not to convince him of the benefits of reality; it is to show him that in trying to escape reality he has only exchanged one set of problems for a worse set and, furthermore, the problems of reality remain. In other words, before introducing classical philosophy, a modern mind must first be convinced of the unsatisfactory nature of whatever modern flavor of philosophy he has adopted.
My experience commenting on philosophy blogs bears this out. I've found there is no point in discussing the virtues of Aristotle or Aquinas immediately, because whatever their virtues might be, the modern thinker usually only knows some of the problems associated with them, and he is usually convinced that he himself is in possession of a philosophy that does not suffer similar flaws; his own philosophy, at best, suffers from minor problems. (On the blogs I haunt, this philosophy is generally some form of empiricism.) Why waste time on some ancient philosophy with unresolved problems when there is a straightforward modern philosophy that suffices? Only when someone sees that the modern philosophies not only have a raft of critical problems of their own, but evade rather than face the problems addressed by classical philosophy, will Aristotle and friends get a hearing.