I think the danger of the public nature of the blockchain is way overstated. It's fairly trivial to introduce reasonable doubt into your transaction history, blockchain analysis is nowhere near as useful to law enforcement as some people think.
The situation I worry about more is when someone is operating outside of the law, for example if you legitimately but unknowingly end up with some coins that were previously stolen, and then some guys with baseball bats turn up at your house.
That's why you split the coins repeatedly as you mix them. You don't want more than 0.1% of someone's "dirty" history. Client-side statistical analysis allows each node to decide which coins are good for them personally. (So that all coins are "far" away from each other.)
The situation I worry about more is when someone is operating outside of the law, for example if you legitimately but unknowingly end up with some coins that were previously stolen, and then some guys with baseball bats turn up at your house.