“The oldest manuscripts we have of the Gospels are only 1000 years old.”
This is a false statement. There are surviving fragments of the Gospels dating back to the 2nd and 3rd centuries:
Wikipedia’s Dating of the Bible: New Testament
“Matthew, Mark, Luke, and John did not sign their names and therefore we cannot confirm that they are the authors of the texts.”
On this point your professor is correct. The Gospels do not claim who their authors are. The names that we know the Gospels by are a product of general Christian tradition. However that doesn’t argue for or against the validity of what is in the Gospels.
“The gospels are not ‘eyewitness’ accounts.”
Police reports and history books are not written by eyewitnesses either. They are reports/summaries of what eye witnesses have said. The Gospels are similar.
However there is a debated opinion that the “beloved disciple” of the the Gospel of John is actually Lazarus. If true, that would make that Gospel written not only by an eye witness but by a recipient of one of Jesus’ miracles.
Let’s put all that aside for a moment. Its not unusual that the Gospels were not written by eye witnesses. Most people is the ancient world could barely read, if at all. The early Christians would have spent most of their time preaching and teaching rather than writing.
“Gospels contain different information and therefore contradict one another.”
This is both a true and false statement. The Gospels are not written as a modern history text book. They are written to inform people who Jesus Christ is and the salvation that He offers. There is great symbolism in the Gospels in addition to historical events. Its a bit disingenuous to criticize the Gospels for not being something they were never intended to be.
Take any historical event. There are historians who take all different views of what happened, why it happened, how it happened, etc. The fact that there are differences of view points doesn’t mean the event didn’t happen.
For more information:
Pontifical Biblical Commission’s
Historicity of the Gospels