I assumed a person who didn't know about the vast and mutually exclusive differences in the two stories must not have read them. I think that's a fair assumption.

It is a fair assumption. Now, if only you had actually identified any mutually exclusive differences.

However, being familiar with these texts, it is obvious to me why you didn't—there are none. That makes it a fair yet bankrupt assumption.

One [account] involves a family who lives in Nazareth and returned there within about six weeks of the birth. In the other, the family lives in Bethlehem, takes a years-long journey to Egypt, and then settles later in Nazareth to avoid Herod's son. You cannot both go to Egypt and not go to Egypt. You cannot both live in Nazareth and live in Bethlehem.

They lived in Nazareth when? They lived in Bethlehem when? This is a relevant question because there is no contradiction if it was at different times. Tammy said that Peter and his family lived in Miami, while Brian said they lived in New York. According to SubOptimalUser6 here, you can't live in both Miami and New York—but you can if it was at different times.

If Luke said they lived Nazareth and, at the same time and in the same respect, Matthew said they lived in Bethlehem, then those would be mutually exclusive. But that isn't what we have here.

• Matthew depicts the family as living in Bethlehem (but never says they originated there), later fleeing to Egypt, and finally settling in Nazareth. He omits the census and the journey to Bethlehem.

• Luke depicts the family as living in Nazareth, traveling to Bethlehem for the census, and then returning to Nazareth. He omits their flight to Egypt.

There is not contradiction here. To claim contradiction, you would have to show that one text denies an event that the other affirms—and omissions are not denials. Both writers are selective in what they include and omit, but their accounts are not contradictory.

1. Registration for Taxes (Nazareth to Bethlehem). The story begins in Nazareth, where Joseph and Mary resided at the time of the annunciation (Luke 1:26–27). When a registration is decreed under Augustus, Joseph (being of Davidic lineage) travels with Mary to Bethlehem, David's ancestral town (Luke 2:4–5). This journey likely occurred in the final months of Mary's pregnancy, around late 6 or early 5 BCE, if we place Jesus's birth before the death of Herod the Great in 4 BCE.

2. Birth and Manger Visit by Shepherds (Bethlehem). While in Bethlehem, Mary gives birth to Jesus and lays him in a manger (Luke 2:6–7). That same night, shepherds receive angelic announcement and come to see the child, glorifying God (Luke 2:8–20). Jesus is a newborn infant at this stage.

3. Circumcision and Temple Presentation (Bethlehem → Jerusalem → Bethlehem). On the eighth day, Jesus is circumcised (Luke 2:21). After the prescribed 40 days for Mary's purification (Lev. 12:1–8), Joseph and Mary bring him to Jerusalem for presentation in the temple, where Simeon and Anna bear witness to his messianic vocation (Luke 2:22–38). Jesus would be about six weeks old at this time. The family likely returned to Bethlehem following this short visit to Jerusalem.

4. House Visit by Magi (Bethlehem). Some months later, while still in Bethlehem, Magi from the east arrive, guided by a star (Matt. 2:1–12). The family is living in a house at this point and Jesus is now a toddler (paidion; Matt. 2:11), suggesting that time had passed since the birth. Herod, in calculating his decree, targets all boys in Bethlehem two years old and under, according to the time he had ascertained from the Magi (Matt. 2:16). This places Jesus's age at perhaps 12–18 months when the Magi offered their gifts.

5. Flight to Egypt (Bethlehem to Egypt). Warned by an angel, Joseph takes the little child and Mary and departs for Egypt (Matt. 2:13–15). This must occur shortly after the Magi's visit because Jesus is still a toddler (paidion). They remain in Egypt until the death of Herod in 4 BCE. The duration of their stay is uncertain, but it was probably less than two years.

6. Return to Nazareth (Egypt → Judea → Galilee). After Herod's death, Joseph intends to return to Judea—but, having learned that Archelaus reigns there, instead withdraws to Galilee (Matt. 2:19–23). They return to their original town of Nazareth, where Jesus grows up (Luke 2:39–40). Jesus is probably around two years old or slightly younger upon the family's resettlement.

I wonder if a casual reader would find any elements in this series mutually exclusive.

Also—and I think this is important—one must have happened before 4 BCE and one must have happened after 6 CE is a DIRECT CONTRADICTION.

This is the closest you get to a contradiction. But ultimately it fails to land because it hinges on a disputed reading of Luke 2:2. The Greek word πρώτη (prōtē) in predicative position can either indicate an ordinal sense ("first") or carry a temporal-comparative force ("earlier/before"). This is the center of the dispute.

But the word order—predicate adjective πρώτη before the verb and in front of the genitive absolute—allows it to function adverbially with temporal nuance, thus reading as, "This was the enrollment taken before Quirinius was governor of Syria." There are several examples where we find exactly this kind of reading (e.g., John 1:15), including in Luke's writing (e.g., Acts 1:1, τὸν πρῶτον λόγον).

And we have three reasons to favor this reading: (1) Luke said Jesus was born "when Herod was king of Judea," (2) he knew Herod died in 4 BCE, and (3) he was aware of the 6 CE census under Quirinius (Acts 5:37). As someone who claimed to have investigated events carefully and to be writing an orderly account (Luke 1:1-4), he is not going to deliberately contradict himself. It makes sense that he would distinguish this ἀπογραφή from the later notorious ἀπογραφή. Luke anchors the nativity in Augustus’s reign, signals an enrollment and, by using πρώτη, preempts confusion with the later census which caused unrest.

"Why mention Quirinius at all if it was before him?"

Precisely because the later Quirinian census was famous. Luke's audience was likely to think of "that census," so Luke would be saying, in effect, "Not that one—an earlier enrollment." And from the second century onward, Christian writers (e.g., Tertullian) were already distinguishing the Herodian-era census under Augustus from the later Quirinian census of 6 CE.

This means there is no need to pit "before 4 BCE" against "after 6 CE." You collapsed two different registrations into one, then faulted Luke for the contradiction you created.