I don’t think Matthew adds these details to make his story more convincing. If anything, they make it more apocalyptic, more hyper-real in the sense of transcending the real. And this is a pattern with Matthew. Just as he ends with many saints being raised after Jesus’ death, he begins with many innocents being killed after Jesus’ birth. Both scenes are completely unique to his gospel.
Similarly, Matthew freely changes details or adds precision not present in his sources. When he tells the story of Jesus and the Syrophoenician woman, he makes her a Canaanite (never mind that Canaanites didn’t exist in this time period). This helps his readers grasp Jesus’ different perspective on what to do with Canaanites and their children.
Other gospels saw a connection to a prophecy when Jesus rides into Jerusalem, but Matthew takes the prophecy hyper-literally. It speaks of a king “riding on a donkey; on a colt, the foal of a donkey” (Zechariah 9:9), which read normally is a single animal described with poetic parallelism. But Matthew alone shows us Jesus riding both animals, somehow straddling both a donkey and its colt as he rides into town. It’s not a detail added to make the account more believable! Instead, it double-underlines with a thick Sharpie how this is the king Zechariah spoke of.
We still do stuff like this today. Movies based on real people’s lives will still take some license in order to arrange events in a way that makes the historical person’s character shine through more clearly. It’s not lying. It’s a way to convey truth that might get obscured in a more prosaic historical treatment.
When it comes to Matthew 5 and the “I have not come to abolish but fulfill” speech, I think Matthew is portraying Jesus as far more in line with Paul than some other gospels. That speech is radical, but it’s easy to miss that if we read it apart from its Jewish context. I got into that more in an old thread here. Or you can see my fuller treatment in an old paper:
I’d say two. John has the raising of Lazarus as an event that leads the Pharisees to further oppose Jesus. Luke includes it in one of Jesus’ parables as something that wouldn’t convince the rich man’s brothers (i.e., “the Pharisees, who were lovers of money,” Luke 16:14). But that’s probably a whole ’nother rabbit trail.