(1) Plato's Republic is still taught in philosophy courses.
(2) Plato's Republic is a book over 2300 years old.
The reason it's supposed to be an embarrassment is that "No other respectable discipline relies on texts that are so old". The contrast is with math and science:
The ancient Greeks wrote about math and science and logic as well, but their writings in these areas are not presented to beginners as valuable instruction; students of those disciplines often learn about them only later when learning about the history of their discipline; and they are often referenced directly only by historians of the field. The only other field I can think of that relies directly on books of such age is religion/theology, and that is not good company to be in, given that we expect philosophy to a) explain the world and b) have rational foundations.
But, of course, philosophy doesn't "rely on" Plato's Republic; it makes use of it. And implicit in the above characterization is an obviously respectable discipline that does make use of texts that old, and even older, namely, history. Peter gives a whole list of possible reasons why we might still use Plato's Republic, arguing that each fails to be a good, or in some cases, an adequate, reason:
1. The Republic is perfect.
2. We are so imperfect in philosophy that we cannot improve on Plato's Republic.
3. There are no philosophical truths.
4. Respect for tradition.
5. Philosophers are idiots, i.e., who misunderstand the purpose of philosophy or believe on inadequate basis 1-4.
6. The questions are still relevant.
7. There is no "official method".
7b. We have no adequate replacement for the Republic.
But in all this is no hint of what would presumably be the most obvious candidate, namely, that one of the things studied in philosophy courses is the history of philosophy. Philosophy is partly historical in nature, and thus has to exhibit it, at least to some degree, from the very beginning. And it is, I take it, very clear why this must be so: a key part of philosophy is being accurate about reasoning, particular the reasoning presented by others. To accomplish this you can't imbibe all your understanding of the reasoning of others from secondary texts. I have met people who were taught on something approximating the method suggested by Peter, that is, who only got their history of philosophy from secondary texts, and they are philosophical illiterates, incapable of understanding any arguments that are not spoonfed to them in terms with which they are already familiar. Nor is it in the least a reasonable argument that math and science do things differently; philosophy is not an ersatz natural science. People who want to play at being scientists should be ignored if they are not actually doing genuine scientific work, because people who play at being scientists, or at being scientist-like, without doing real science, are in general dabblers in quackery. If you ask me, the real embarrassment in philosophy is that you can find people who play at being scientists in this way. It's like people pretending to be doctors in order to be taken seriously; they'd be easier to take seriously if they just dropped the pretense.
But that's a digression. To return to the more immediate point: If there are people who teach the Republic "outside of the context of understanding the history of the field, or an understanding of ancient Greek literature," then the natural explanation is that they are incompetent teachers of philosophy. If there really are such people, I agree that it is a shame that they are allowed to teach the subject. But if they do exist, they cannot be all that common, because it seems quite clear that most people who teach the Republic do not teach it in this way; rather, they teach it as a taste, a first introduction to the riches of the history of the field. And it is clear that a work like the Republic is quite good for this sort of thing, for several reasons.
(1) It has been immensely influential through history, and therefore it's a useful text to start familiarizing students with, because they'll come back to it in one way or another, through reference, or allusion, or kindred argument.
(2) It is readable; indeed, philosophically inclined students often enjoy it immensely. It is also very discussable, as years upon years of introductory courses have shown. Both of these are essential to an introductory philosophical text.
(3) It has a mixture of both the strange and the familiar, and thus provides an excellent opportunity to start helping students sort out the difficulties of reasoning that takes place in terms with which they are unfamiliar. That is, it's a good text, and has through time been shown to be a good text, for starting students on the work of honing their analytic tools.
(4) The themes it discusses are big ones -- justice, death, knowledge -- and thus it allows for considerable branching out. This means, among other things, that the text allows for considerable pedagogical flexibility (which is a big plus in an introductory text) and that it provides a way to get students interested in other arguments and reasoning on these themes, whether ancient or modern.
One could, of course, choose other texts, from just about any time period, on similar grounds. That will chiefly be a matter of taste; but there will be works whose use exhibits especially good taste, in the Humean sense, and the Republic is certainly one of them.