Not exactly like cash. More like a bank, because it is traceable. You can receive regular payments to your bank account for innocently delivering pizzas to a drug cartel. But getting money from a drug cartel will raise suspicion. Of course you can prove you were just delivering pizza, but the aim is not even to have to prove it.
With cash it indeed doesn't matter who you receive it from. But if with it didn't matter then you could just create a new address, send the coin to that address and claim you bought those coins - no need for mixing, right? But people still use mixers so..
Two things you should add to your service:
If you choose 2 forwarding addresses and the difference is 1 hour, it will be exactly 3600 seconds. You should make a feature so that the time is random within a certain range.
For a larger fee you could give out freshly mined coins. You could probably buy new coins from miners.