Oh dear, you contradicted yourself. The note starts with the declaration: “Oedipus Rex” is a tragedy of fate. This was what drew my attention to the post, for I disagreed with that sentence. Later on, you offer the corrective yourself: If Oedipus is the innocent victim of inescapable doom, he would be a mere puppet and the play becomes a tragedy of destiny which denies human freedom. Sophocles does not want to regard Oedipus as a puppet.

Here's a way out of the confusion: is Oedipus Rex the play a tragedy of fate? No, as you show toward the end of your post, Sophocles is concerned with showing Oedipus as a free agent. Bernard Knox in his introduction to the Fagles translation gives us a reason for Sophocles' motivation: he lived in an age of skepticism much like ours (an age exhibiting the "failure of piety" you point out), where ancient prophecies and the authority of gods were under attack.

Now, is the myth of Oedipus a tragedy of fate? Most certainly, yes! Let's remind ourselves that Sophocles' play does NOT concern itself with the events of the myth that take place before Oedipus goes about his business of ridding Thebes from the plague. So your first three paragraphs are not, strictly speaking, about the play, but about the myth. True, the play does depend on the myth, but if you dwell on the myth too much, you end up abandoning Sophocles' vision. I notice the bias in your answer. Let me explain. There is nothing in the play that condemns Oedipus for being rash and arbitrary, either for his murder of Laius' (in the play, the only testimony is Oedipus', and he says that he merely "retaliated in good measure"), or for his behaviour with Creon and Tiresias. You also said "easily provoked", which couldn't be further from the truth. Think about it: you take it upon yourself to avenge a murder, and someone tells you, in a public gathering, that you yourself are the murderer you seek! His frustration may be a little too expressive, but it is certainly not out of line!