Exactly, why is it not incumbent upon them to prove that you have gained that money through illegitimate means?
If I open a civil proceeding against you, I can try to prove that you owe me $5k and if I prove that to the appropriate standard you can be compelled to provide me with $5k. I can't hack your bank account, withdraw $5k, and then open a civil proceeding to retroactively legitimize my robbery.
The whole system is a massive violation of due process, no matter what legalese you try to use to justify it. If you're allowed to take people's property and then retroactively sue them if they complain, what is even the point of having the Fifth Amendment?
If there is reason to believe that someone is in possession of property that doesn't belong to them it seems perfectly reasonable for the police to confiscate said property.
That's not what's being described here.
If it turns out the guy running away from the bank just likes jogging with a bag of money the police should return his money and whatever else _immediately_ after the person is released.
How about "police should not take property where no crime is known to have been committed"
Yes. The money may be held in escrow pending the trial, but appropration by police is unjust -- and should be illegal if the person is not charged and arrested.
Edit: all of this is off-topic, as the bank can be proven the owner and the police would not keep the money.
Asset forfeiture is, and always has been, subject to judicial oversight.
There are, in fact, well established and relatively simple procedures for contesting civil forfeiture. There is a fair argument to be made that they sometimes don't work as well as they should but when you say that there is "NO judicial oversight" at all you are just just flat out wrong. You need to update your worldview if you want it to conform to reality.
(Edited to add: I just realized they didn't actually get their money back. They got a check. Yep. You can generally cash those for free if you have a bank account.)
Imagine the situation of someone whose circumstances are less clear, or who can't otherwise afford to stay in one place and fight the seizure for however long it actually takes.
What happens to the person who just had all of their money taken away?