Actually I'm in UK although it might be the same here. Second point still stands though: you can't tell if someone has been into your house.
Edit: to elaborate on "the silverware", "the police" having access means effectively anyone has access. It's not just 1 infallible person, you're giving that key to the whole station, and it's being kept somewhere, which makes it (more) vulnerable to theft, especially since a large cache of such keys is going to be much more attractive to a criminal than a single key.
It's actually a myth police have to have a warrant in the US. There are a couple other options
1) Hot pursuit
2) Exigency
Domestic violence falls under the second. Battering ram can come out.