How do mixers work? Do they use system similar to P2P if the number of coins of multiple users match each-other? Then it can create serious problems.
I know mixing is not money laundering but look at this:
1. Mixer receives money from someone who has done illegal activity.
2. Mixer stores that money on address A
3. I create a mixing order, deposit money to mixer and hours later I declare address B where I want to receive money.
4. Mixer sends me money from address A to address B. Address A has received money from someone who has done illegal activity, so there is a chain: Laundered money -> Address A -> Address B that belongs to me. This is probably what exchanges see and it's very hard to convince them that you are not associated with illegal activities.
Mixers can utilize coinjoin as well as doing several transaction with new addresses in the outputs to span out their transaction tree and make it harder for detection software and/or sleuths to locate fund origin. Of course this is expensive and many mixers will just do what you describe without and steps in the middle.
At it's most basic, a mixer will pay you with someone else's money just so the trace of your initial BTC ends up going somewhere else. But if a mixer is only doing the latter of the afformentioned however they run into the risk of their users receiving tainted coins which can't at all be used in many services as with OP and might end up getting whoever deals with these coins watched or suspended... However, even with the techniques I mentioned at first, it's still possible for the more sophisticated software to detect fund origin.