Why is it important to encounter the choragos throughout the entire scene?

As you haven't mentioned which particular scene you are referring to, I'll address how the choragos functions in the play as a whole and why his presence is significant.

The choragos is the leader of the Chorus, which consists of the elders of the city of Thebes. One feature of this chorus that is distinctly characteristic of Sophocles is that they are actually part of the debates in the play, rather than just performing odes...

As you haven't mentioned which particular scene you are referring to, I'll address how the choragos functions in the play as a whole and why his presence is significant.

The choragos is the leader of the Chorus, which consists of the elders of the city of Thebes. One feature of this chorus that is distinctly characteristic of Sophocles is that they are actually part of the debates in the play, rather than just performing odes in which they pray or reflect on the nature of life and fate.

The choragos is the only member of the chorus who speaks individually rather than dancing and singing as part of the group, and therefore his presence allows Sophocles to give voice to the opinions and view of the general populace.

The most interesting scene in terms of the role of the choragos begins at line 320, when he has a dialogue with Oedipus. The first thing the dialogue shows is that Oedipus is a good ruler rather than a tyrant; instead of simply giving commands, he asks for the opinions of his subjects and takes them seriously. We can see this in the following exchange:

OEDIPUS

You may indeed [make a suggestion], and if there is a third course, too, don’t hesitate to let me know.

CHORUS LEADER

Our lord Teiresias, I know, can see into things, like lord Apollo.

The choragos remains on stage to witness the words of the blind seer for two reasons.

First, there is no need for him to exit and he'll be leading a choral ode immediately after the dialogue; thus having him exit would be unnecessary and distracting to the audience. Next, his remaining on stage shows that the citizens of Thebes are active participants in the quest to locate the murderer, again showing Oedipus' leadership qualities.