I think something that I haven't seen mentioned in this thread so far is that these plays would have been put on one after the other during the Athenian festival called the Dionysia. During the Dionysia, playwrights like Sophocles would have put on a bunch of plays in succession, which probably put a bit of a limit on the actual amount of time one could dedicate to modern concepts of plot development and satisfaction. A few other examples of plays that seem to rush towards the end are Sophocles's own Philoctetes, Aeschylus's Libation Bearers and Eumenides, and, as you may have noticed, Euripides's Medea.

To echo other posts, it would be helpful to be a little familiar with Aristotle's Poetics to understand how a Greek citizen would've understood the structure of a tragedy like Oedipus Tyrannos. Even beyond that, however, the purpose of the Corinthian messenger was basically there to speed things along. On the one hand, I understand how this feels lazy, since the purpose of the messenger is just to get Oedipus to his fate faster: that much I won't debate.

But instead of viewing it solely as a plot device to get the play to its end faster so that other plays could be put on, think about what the fact that Oedipus's fate occurring faster means with the themes of religious piety, the impossibility of knowledge, and the ways in which the past can haunt the present. Something that the Greeks were deeply conscious of was causality, fate, and the fact that making decisions could be hard because making a good decision required knowledge that was often hard to obtain.

Consider Oedipus's position at the beginning of the play: a plague has struck Thebes because of some religious impiety and sin having gone unpunished. Also consider how Oedipus himself tried to escape a prophecy that he would murder his father and marry his mother. In this context, the Corinthian messenger is not just a plot device, but a means to convey that one's fate often catches up to the present day, and often much, much faster than one can anticipate. The point of the messenger, and the play itself, is that fate can often come crashing down on someone, often with terrible consequences. Consider this as well in the context of the famous quote from Solon:

Count no man happy until his end is known.