>>717154754Because he confessed to the murders and explained exactly how they happened. However, what you said is correct, "death by TV World" is unprovable. Hence despite Adachi confessing, providing a motive and authentic time to which he did it, the police are having a really hard time making the charges stick. That's why he's still in jail a year later in Persona 4 Arena, awaiting a sentence.
And Adachi confessed because Yu beat him. Not in battle, but in the game he was playing. Adachi viewed the whole murder mystery as a joyride fun game to kill his boredom cause nothing ever happened in Inaba. He set up Namatame as the fall guy and idiot to continue his work, and when it looked like the game was about to end when Mitsuo killed Morooka and took credit for all the murders Adachi kidnapped him to keep the game going.
Yu successfully deduced the true culprit and defeated him, so Adachi gave up the ghost. His speech about society was just a deflection.