[cornholio]

Ah, I see, the real ChipMixer had a major flaw. I had no idea how the entire system operated, I used it just as an example to illustrate off-chain custody handovers. My "logically impossible" scenario was that the mixer has an array of addresses on the chain funded by previous customers, and when a new customer comes in it just runs a knapsack on that set and assigns them a subset of keys. Perhaps add a single layer of coinjoins to dilute each "really bad" incoming transaction, so clients won't directly get the bitcoins laundered by kidnappers and it's transparent to the whoever is doing the tracking that the coins have been laundered.

> why would having a private key to an output address that no one else has touched be an evidence to a crime?

An address is a hash over an ECDSA public key and a public key is a computational derivation of the random private key. If you have the private key, you can derive the associated address which is publicly connected on the blockchain to known proceeds of crime that have been laundered. That they were spent or not (by an another customer than the criminal) is irrelevant, it proves that you handled them.