Why couldn't Kris Kringle....

Like Jesus, who could have done some miracles to prove who he was and save himself, Kris believed he needed to be a martyr in order to get across the point that the world was going to hell in a wheelbarrow as far as Christmas was concerned.

In addition to what liscarkat said, he wanted to defend the spirit of Christmas, a large part of which for children is the magic of it, the faith in it. A big thing in this movie was him teaching Susan to have imagination and believe in a world bigger than what she could see with her eyes. Producing his reindeer in the courtroom and turning it into a matter of hard evidence would defeat the purpose of this and take the magic out of it. It would make Christmas more mundane, taking away a lot of its power. He wanted people to have faith. And he felt that if people weren't having faith anymore anyway, then it wasn't worth trying anymore.