First of all, all of the gospels are anonymous. It was not the nature of the genre to include one's name in the body of work, so they are all anonymous.
Second, though, is that in the case of all four of them, there is no evidence from the earliest of sources that their authorship was ever in doubt. Every single reference to them in early works agree that the four authors were indeed the authors. There is no disagreement anywhere about this. As a matter of fact, if authorship was going to be contrived, other names would have made more sense, viz. Peter, for one.
Thirdly, to be the only one to record an event doesn't give evidence that it's untrue. Think back to Watergate. The story was broken by Woodward and Bernstein, and they were the only ones telling that story. It didn't make it untrue.
You say "Luke is hearsay." Interesting. Luke is the one who says specifically that he "carefully investigated" to write "an orderly account" so that these things might be known "for certainty." That sounds like research, not rumor, to me.
Though Mark was not one of the 12, it is strongly thought that he was a follower of Jesus and travelled with him (
http://en.wikipedia.org/wiki/Mark_the_Evangelist). It is even thought by many that he was in the Garden of Gethsemane on the night of Jesus' betrayal (Mark 14.51-52).
Why do I trust the accounts of the four gospels?
1. More than any other documents in ancient history, we have hundreds of thousands of pieces of the NT, enough to give us an incredibly reliable text to work from.
2. The gospels show historical accuracy.
3. They show archaeological accuracy.
4. Most estimates are that they were written within the lifetime of the eyewitnesses, rendering them critique-able and verifiable.
5. The accounts are filled with details that could only be known (or would bother to be recorded) by eye-witnesses, giving the impression that the originators of the stories were actually there.
6. The four accounts, though different, are clearly complementary.
7. The oral culture of the day gives us adequate reason to believe the stories wee communicated and preserved without the necessity of exact wording. (It couldn't have been anyway, since Jesus spoke in Aramaic, and the gospels were written in Greek.)
8. The writers of the gospels were not above criticizing themselves, a mark of both honesty and integrity.
9. The writers of the gospels are known by history as men of integrity.
10. Based on what we know of their lives, the era, and their accounts, they can be considered as credible sources for the information written.
11. There is no hint of any kind of conspiracy, fraud, or collusion.
12. The events they wrote about were public occurrences, not private experiences. Detecting fraud or error would have been easy.
13. There is no hint that these men were insane.
14. Their writings have been pored over for millennia and have been accepted by great numbers of scholars.
15. While they certainly lived in a non-literate culture, they didn't live in an illiterate one. These people were obviously not buffoons, barbarians, or blockheads, but were demonstrably fairly cogent and logical.
Just naming a few.