Often the actual laundering process is done by a third party who takes a cut. A middleman who provides money laundering as a service to criminals. Under your scheme that person wouldn't be committing a crime and could not be prosecuted.

But that's beside the point. The crime isn't "moving money around in unconventional ways," it's lying to the IRS and other government agencies. It's signing your name on all those tax documents that say "under penalty of perjury" after filling them in with false information.

If someone commits perjury, convict them of perjury. There's no need to invent yet another crime. But I suspect it's not perjury. If a laundromat has an extra $500k of income and reports it as income, paying all relevant taxes, then so be it. If there is proof that the money came from criminal activity, then convict someone of that crime. Otherwise it's just a hunch.

One of the purposes of making money laundering a crime is that it makes it possible to investigate other crimes. You know the mob is making money and you know they're moving that money around somehow. Making money laundering a crime gives investigators a place to start their investigations. If they think a laundromat is a front for the mob, they can prove that by counting how many people are using it and comparing that to how much money they're putting in the bank at the end of the month. They can then get a warrant to investigate further. A judge shouldn't issue a warrant on a laundromat for investigating a murder when there's no clear thread that connects the two. A judge could however issue a warrant to investigate money laundering when you know the laundromat's books don't add up.

And thusly we went full-circle back to the OP:

Now LE comes along and says, gee whiz, our jobs are really hard having to catch criminals doing criminal stuff. You know, let's make it illegal to use your money as you like, to have any sort of privacy in your transactions. Tada, now they can just go after people for moving money around.

Investigating organized crime would be significantly harder without being able to follow a paper trail. I'd rather have the police be able to get legitimate warrants to investigate actually suspicious activity than have them operate by illegal and unethical means.

Making it a crime serves as a deterrent for illegal activity. So when you prove the money came from criminal activity you can convict them for the criminal activity and the money laundering.

This seems similar to how we punish some crimes worse if you thought something mean about the person you committed the crime against.