Why should they be careless with either of them, any service that holds customer data and funds has to ensure it is securely stored. Ledger lost their customer database many years ago, and till date those customers are still attacked through phishing emails. If you are going to collect data, ensure it does not leak, if you are holding funds, ensure you don't lose it to a hack, because the both are important and i am not picking one over the other.
In the terms and policies of some centralized exchange, they state that they don't accept Bitcoins from an untrusted source such as mixers or a wallet that has been flagged, some exchange even state that they don't accept Bitcoins from casinos.
So, if any customer disobeys those rules and send Bitcoin from the mixer on any sources that the exchange doesn't want, they will block the persons account and the customer could lose their coins.
The exchange are doing so to protect their self too from government policies, I guess.
So, when they cease those coins, how sure are we that they will take it to the government?
Sometimes, after making such deposit, before the exchange detects it, you must have traded the Bitcoin to Fiat and withdraw it to your account, so what will the exchange do after you have already withdrawn the money to your account before they later found out that the coins was tainted?