I'm tempted to write "Why study Judaism?" and leave it at that. Rabbi Jeff Holman's answer above concisely articulates that without the oral/rabbinic tradition recorded in the Talmudh, most extensively and clearly in the Gemara, we literally do not have any basis for "our" Jewish beliefs as distinct from Qara'im, Seduqim, Ebionite Christians (I specify here because they kept Torah law and did not accept the divinity of Jesus; merely his prophecy)...
I don't fully understand the implication of the question, that is to say, the notion that there's an alternative to students in a Jewish institutional setting (generally spending years, not to mention truckloads of money, therein) learning the very foundations of their religion. Would anyone asking the question seriously not assume that Talmudh provides the foundations of Jewish religion, along with torah shebikhtav?
Otherwise, one might similarly ask, why teach Tanakh? Why use Jewish texts at all?
Assuming that literacy in foundational texts is valued, and assuming that we believe the Rabbinic (as opposed to, say, Qara'i) tradition carried and carries the Oral Torah from generation to generation, what's to debate? Time budgeting? How many other parts of a Jewish Studies curriculum can take precedence?
In my opinion, the more urgent questions follow a thread like this: Why do such a preponderance of Modern Orthodox-educated students and alumni (for the example most familiar to me, though other types of Jewish day schools seem to fall similarly short in this regard) not know how to learn Gemara, maybe even Mishnah, on their own? Should we accept that only a minority of students will ever understand the elementary building blocks of what becomes halakhah and hashqaphah? If not, how can we close the gap between goals and achievement?
