500 BC Mencius, Lao Tzu, Buddha, Confucius

Could Jesus's teaching have been influenced by
folks who went before, in his own and other lands?

Certainly it was important for him to speak a language understood in His time ( I capitalize His in a similar way I capitalize U.S. Constitution, which also is subject to interpretation ).

Certainly it was necessary to use phrases and idioms understandable to persons living in His time.

Certainly He must have discussed matters of importance to human beings rather than to giant gas bag beings on the planet Beemp.

What "spurred" Buddha ? Goodness I suppose.
Did Buddha's story reach Jesus' culture. I expect it did. I think the story of Jesus goes beyond Buddha's. Our Prince Gautama was born wealthy and enjoyed a marriage before voluntarily entering a life of moderate austerity and teaching. His times and country were not invaded by a powerful, somewhat atheist military government. He died an old man after a nice meal.
Jesus' story puts the son of man in a more difficult birth and life, and ends the way so many have ended...tragically. But the message is that tragedy is being conquered on the other side and that victory is within your reach.
I'm not a theological writer, just summarizing.

Supposedly, Jesus said He was the son of man....and he also referred to himself as Mary's son. According to the story, the Pharisees felt he performed in ways which suggested He was either the son of God or was referring to himself as such. I can accept the story as a son of man who somehow seems to be operating from another realm...one from which the universe came.
Why should it not be possible? And in the story, He has marked a path, a doorway...subject to interpretation, of course.