Recently I was listening to a sermon and the preacher said that "John was written for Christians" among some other statements about who the four gospels were directed at. Very often I hear statements that Gospel X was directed at: Jews/Gentiles, believers/non-believers, people at Rome/people in Judea/etc..
This is not a bad question to ask. We ought to be very interested in understanding who the gospel documents were going to be read by, it helps us understand what they are saying. But the answers are often very poor.
Reading recently something by E.A. Judge I was struck by lightning - all 4 canonical gospels are in Greek. That means every single one of them is written for a Hellenised/Hellenistic audience. The choice of language determines that. None of them are aimed at Galileean peasantry. None of them are aimed primarily at Aramaic speakers, unless the target Aramaic speakers are bilingual in Greek. This is especially important for Matthew. However 'Jewish' one decides Matthew is, it's written for Greek-speakers.
Secondly, some of these are false dichotomies, or gross oversimplifications. In my view none of the four are more 'Jewish'. All of them make extensive use of the Old Testament, albeit in different ways. Nor would I be inclined to say any of the gospels was exclusively aimed at insiders or outsiders. Luke may be presenting more with attention to outsiders, but he is also so steeped in OT that he clearly has insiders in view as well. John, which my preacher said was "written for Christians", tells you explicitly and clearly that he wrote his gospel to engender faith in his readers! It's an evangelistic book through and through. It's also very clear that John is not 'only' for outsiders.
I don't think these questions are going away. But I wish stupid and facile answers to them would.