It seems pretty clear that both Matthew and Luke intended to replace the gospel of Mark, and Matthew almost succeeded in this; it is less certain that Luke knew Matthew’s gospel or vice-versa, ‘though I do think Luke at least knew of some of the gospel of Matthew. It is also not at all certain that the author of the gospel of John wanted to replace the synoptic gospels.

On a functional level Matthew did replace Mark, so much so that the early church fathers didn’t really know what to do with it. They settled on the explanation that Mark was some sort of synopsis of Matthew, acknowledging implicitly the preeminence of Matthew. For me this reinforces the idea of Markan priority. Mark really makes no sense in relation to the other gospels as other than the first. What I’ve always wondered is, if Mark had been entirely lost, could scholars deduce its existence simply from Matthew and Luke? Or would scholars simply assume one of the two copied the other?

All the folks I’ve read who think John knew Mark seem to assume it was intended as some sort of supplement to Mark, if not actually a direct response to it. I’m completely agnostic on this. That’s why I look forward to Mark Goodacre’s promised tome on the question. I can be convinced by a good argument.

Here’s a wild hair to pluck. Might it be best to think of Matthew and Luke, not as independent gospels, but as different versions of Mark?
Matthew contains over 90% of Mark and hypothetical “proto-Luke” mirrors Mark very closely (while also containing over half the book). Perhaps what we have here are not independent literary expressions but a single literary expression. Then only John, even if it responds to Mark in some way, did not begin as a version of it.