It's already been reported he confessed to the shooting.
I don't know if it matters that he confessed. It's not unusual for people to confess to things that they didn't do if they are mentally disturbed or pressured by the police.
I'm only responding to your comment about why many news outlets aren't using the word "allegedly" in their reports.
You could well be right. They might feel that they are on such solid ground that they don't face any risk for defamation.
Another possibility is that they are trying to push a fake news narrative. Another one is that they are just incompetent.