How can the gospels use omniscient narration?

I think there are two lines of defense.

First, the authors weren't writing in a vacuum. It's entirely possible that they spoke with people that would have had first-hand knowledge. While this doesn't, as you note, necessarily guarantee word-for-word accuracy of any recounting, it would suggest that we might expect at least the accuracy of any other historical retelling of past events.

Second, scripture is inspired by God (n.b. 2 Timothy 3:16, also Biblical inspiration and Biblical inerrancy), who, being omniscient, certainly would know what happened, even to the extent of verbatim accounts of what was said.

On a related note, the quote mentions omniscient narration with respect to bias, i.e. an increased lack thereof. In fact, when you read through the bible, one thing that stands out is how often it records events that are, shall we say, less than flattering. I wouldn't call the Bible "unbiased", but the bias is primarily God's bias, rather than that of the authors, who are generally reporting what happened in an honest and frank manner. (Particular examples include the disciples running away at the Crucifixion and the empty tomb being initially discovered by women.)

The "omniscient narrator" style is not unique to the Gospels/Acts; it is abundant in ancient Roman, Greek, and Jewish writing (e.g. Tacitus, Dio Cassius, Thucydides, Josephus, all over the Old Testament, etc.).

The use of this style does not automatically disqualify the reliability of the text, but rather, leads to the question a historian should be asking of any text: can I trust this author?

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Reliable authors

The reliability of the Gospel authors is an immense subject on its own, and one's conclusions are easily swayed by pre-existing beliefs.

• For a brief, early defense of the reliability of the Gospel authors, see Irenaeus of Lyons Against Heresies 3.3.1, showing that the people who authored these texts were in a position to know what they were talking about.
• For an extensive discussion from the standpoint of a historian, my video series here may be of interest. Among other things, I examine why it is rational to believe in the reliability of several New Testament authors, regardless of one's theological leanings.

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Specific Examples

This post on the Hermeneutics site walks through 5 such examples and may be of interest.

Mark and people's inner feelings

We have multiple, early historical attestation that the Gospel of Mark was based upon the teachings of Peter (e.g. Papias, Clement of Alexandria, see a more extensive discussion in the video series linked above).

If true, this means we're seeing people's reactions through the observations of someone who was there and knew these people personally. Though Peter surely did not know their every thought, powerful emotions can be readily noticed by an observer.

Re The narrator explains why characters do things, we can simply rephrase this as The narrator explains his understanding of why characters do things, in which case this question largely collapses into the question on the reliability of the author, addressed above.

Pharisees thinking evil in their hearts

Our source for knowing what the Pharisees were thinking in this case is Jesus Himself. If the rest of the text gives us reason to believe Jesus is a reliable source and these are His words, there is no obstacle here.

Pilate knew they were envious

The hostile politics between Pilate and the Jewish leaders was general knowledge, as was the Sanhedrin's envy of Jesus' popularity with the people.

Herod's conversation with the Magi

Although Matthew probably didn't get this information from Herod, it would not have been difficult to get the story from the Magi or their associates. Do we really think the Magi, recognizing who Jesus was, just disappeared after their trip to Judea and never said or did anything about it ever again?

Demetrius

His ill intent against Paul became publicly known when they leveled accusations against Paul before the town clerk a few verses later.

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Conclusion

If we are simply trying to answer the modest, historical question of whether the events reported in the Gospels/Acts could have been known and reliably reported by the authors, the answer is a resounding yes.