It was completely stupid they let him at the controls on the bridge. They should not have just continued to trust him. If he's a whole new person then he's a whole new person that needs to earn people's trust, maybe pass a few exams to make sure a chunk of starfleet training didn't get lost in Neelix's recipe index.
As to killing the EMH, that might have happened if it had gone on a few more weeks. I think he would have outgrown his Neelix insecurities and possibly become ruthless, but he was kind of an embryonic personality still when they came for him.