Only because by that stage Julie had known how long he had been planning them.
Jeremy frames himself for the crime by telling Julie that he had indeed carried out a crime he had previously told her he had been planning.
Why not just not tell her he carried out the crime and let her and others suspect him without proof?
None of this makes sense.
Why didn't Julie mention the silencer, something that, interestingly enough, an innocent Jeremy wouldn't mention? The silencer was a major issue during the killings, assuming it was used. It's almost unbelievable that it wouldn't be mentioned.
Bear in mind, also, that if Jeremy had returned the silencer to the gun cupboard in order to allay suspicions about it being missed, Jeremy also knew (or at least, could reasonably infer) that the police had not found the silencer, or if they had found it, they had not understood its significance.
This means Jeremy was home and dry. Why not just take the keys off Dozy Stan, thank him, invite him in for a whisky or three, send him on his way, then dispose of the silencer?! Simple.