You are talking about dirty/clean coins and taint which is not something we as bitcoiners should accept as any kind of standard.
While true, I think what actually matters is how you want to use the coins after mixing. If you want to send them to an exchange, then you should probably check the history before just in case. Not doing so just because of your beliefs will probably cause you trouble in the long run, it's the same as wanting to become a doctor, but then you don't study for the local university's admission test because you don't agree with the subjects.
Here is a simple comparison to fiat alternatives.
If I know your PayPal address, I have no idea how much money you have in your account. I don't know your connected bank account/debit card or the history of your account.
I also know nothing about your account balance if I know your IBAN number. I would have to work for PayPal or the corresponding bank to have more information about you and your money. That means that everything is private between you and your financial service providers.
But Bitcoin is a public ledger. If I know your BTC address, I can see how many coins you have. I can see when the coins were deposited into your address and from where. If you aren't careful with coin control and you reuse addresses or consolidate funds, I can find out a lot about you and your wallet. That's not something you want people to know.
A mixer breaks that link. It's like a bingo cage. Everyone puts their balls into the machine, it spins, and each participant gets a different ball. You wipe out the history of your coins and inherit someone else's.
I've always been curious to know. What if inheriting someone else's history creates trouble? As a result of covering the tracks in the mixer, you will get someone else's thread with a dark history that will lead to you. Even despite the fact that you are not personally involved and have nothing to do with it.
Is this even possible?
It depends what kind of mixer you are using. If you use one that has "mixing code", then what this means is that you will get someone else's coins and automatically inherit their history. This makes it possible that you could receive coins straight from a major hack/ransom/ any other illegal activity that you don't want anything to do with.
Using Whirlwind this can't happen as we pool
all funds together in a multi-sig, the same multi-sig that is also used for withdrawals. Instead of getting "someone's" coins, you get coins that
could have originated from any deposit into Whirlwind, so you can't be directly linked to anything. The privacy set (amount of deposits from where your mixed coins could have originated from) will become stronger with every executed transaction, so in a very short amount of time after we get some usage we will offer by far the strongest privacy of any mixer on the market.