What do you expect the police to do in this situation? They can't just walk into people's homes over your word. OP at least had a noise which in the states I assume is probable cause or something. And even then they'd have to come back with a warrant but got the headphones because the suspect felt defeated.
Your situation is bizarre, the helpless feeling of knowing where your stolen item is and not be able to do anything about it. But it's not solvable by the police unless you have more evidence. For example you could have recorded your neighbor saying those things to you.
Here in Sweden petty theft is 99% an insurance issue. I even heard one case where their boat was stolen with an airtag on it and it was traced to a particular driveway.
The police here won't even pick up the case, it goes into a pile of other theft cases and the owner can use the ID to file an insurance claim.
It's very sad on the surface. But maybe they have their reasons.