If you were actually advancing an argument that academia is useful because it publishes research, you need to prove that the research does enough good to justify the costs it extracts from it's students.

Ok, there's a confusion here I feel a need to correct: research is almost entirely not funded by students. Teaching is funded by students. Administration is (gratuitously and copiously, beyond anything necessary) funded by students. Teaching and administration are also often funded by endowments and state block grants. Research is (by and large) funded by research grants, and in fact, the level of research output required to justify each dollar of grant has gone solidly up.