> When “doing crime”, I’m really not interested in committing the transaction to a permanent public ledger.

As the recipient of the funds you need everyone else to agree that the funds are actually under your control. That requires a public ledger associating unspent funds with access credentials. The payer needs the same thing so that the recipient will accept the payment. ("Permanent" goes without saying; even if old transactions can be pruned from the database there is nothing to prevent someone from keeping a copy of the full history.)

What you want to avoid is having your real-world identity associated with the funds, which is where privacy-focused coins like Monero come in.

Even if you handle payments offline with briefcases full of unmarked bills you're still leaving a trace an investigator could follow, possibly even more easily than they could follow a trail of transactions through a suitably resilient cryptocurrency.

Doing crimes involves making many risky actions which the criminal isnt really into. It might just be that making the crime using cryptocurrency might be seen less risky as doing it with some other tool.