"Innocent until proven guilty" is very different from "Guilty because most people in similar situations are guilty, so let's not bother to prove it in this case."
The cops would have a field day, for instance by arresting college students at random on the weekend because statistically most will test positive for controlled substances.
The concept of "proof of innocence" does not appear in the Constitution; it derives from common law, and the Latin word it comes from doesn't mean "prove" in the mathematical sense.
Which makes sense, because "innocent until formally proven guilty" doesn't make sense in a sentence that proceeds to "beyond a reasonable doubt".
That's an interesting question, but it's not obviously greater than 50%. Honestly, I don't think it's higher than 50% at all, but certainly it isn't so clearly higher that it should be accepted without evidence.
I'm not a conspiracy nut but I can certainly see how someone who is could view this as an effort to legally coerce citizens into using only traceable methods of payment approved by commercial processors under the implication that something like cash or bitcoin would only be used by those trying to hide something.
Think about it statistically. The standard is "more likely than not" the money is obtained illegally. If you pull over 100 people with large amounts of cash, what percentage obtained it illegally? If it's greater than 50%, then you can always seize the cash under that standard. You don't need to make the judgment that no one has a legitimate reason to carry that cash, just the weaker judgment that it's statistically unlikely that the cash is legitimate.