As far as I know the scholarly consensus is that none of the gospels, including Matthew and John, were written by disciples, or anyone who lived within Jesus' lifetime. Obviously Christians believe otherwise.
* the gospels were written in the 1st century
It is therefore entirely possible that they were written by eyewitnesses, even though many do not think they were written by some of the 12 disciples. The topic of 'eyewitnesses' is however hotly debated. See e.g. https://www.amazon.co.uk/Jesus-Eyewitnesses-Gospels-Eyewitne... which is pro this view but also plenty against.
Even John's gospel, which is often thought of as the latest, may well have been written very early; arguments for a late dating are almost wholly made in relation to the text itself (i.e. it has a 'higher' Christology) and not wider historical data.
Source: I am studying theology at Cambridge University in the UK and have heard several professors here debate these topics, plus I am familiar with the literature.
Some of the teachings of Jesus might be historical.
Depends on the Christians. My Catholic school teachers in Germany taught us what you write.
The "cool" youth pastor who was responsible for these events told us "the Gospel's authors are anonymous, their names are totally traditional". I never had the sense that this view was in any way heretical or contentious, even in a strain of Christianity that strongly emphasized the historicity of the Bible.
There's been this weird push to view the Bible like the Quran and the two really have nothing in common. The entire view on the book is wholly different.
The authorship of the Bible is actually not really important if you believe the claim of the Catholic/Orthodox church (who make the same claim)