We are talking about a time before the invention of the printing press. Thus writing at that time was very different from what it is today.
Satire is usually written to amuse and entertain. Today you can achieve that by writing a book of satire, and having it published. But at the time of Matthew, that wouldn't be guaranteed to get you an audience.
My guess is that a satirist at that time would have written his satire either in the form of drama, or in the form of dramatic speeches to be performed at various places. So it would probably be written in a more dramatized form than contemporary satire.
Maybe I am mistaken in these guesses. Either way, there is a question of whether those considering Matthew to be satire are taking into account the way that satire would have been written in that period.