Why is Luigi pleading not guilty? This doesn't make sense

This isn't adding up to me.
Luigi Mangione's lawyer is saying that he will be pleading not guilty and that he has seen zero evidence that he is the shooter. Why would Luigi carry a manifesto admitting to the crime while lounging at a McDonald's, like he was begging to get caught, and then proceed to plead not guilty? To add to this: He allegedly disappeared for 3 months before the shooting. Where was he and what was he doing during this time? Who was with or around him? We need answers to these questions... This seems extremely odd. If he wanted to be caught to show his demented message to the world, why say you didn't do it in a court of law?