First of all, there's no thing such as absolute privacy or anonymity. So while there are ways which try to break link between A1 and other Bitcoin address you own (i'll call it A2), there's always possibility someone can make link between A2 and A1. For example, amount sent from A1 and amount received by A2 is extremely close within short time interval.
The work of blockchain analysis revolves mostly around guesswork and playing the "this is more probable card." If you move BTC from a KYC-ed exchange, there is no doubt that the address you transferred the coins to is somehow connected to you. Now, it doesn't have to be your address. You might have sent the coins to someone else or paid for a service. But what's important is that they (the exchange and blockchain analysis) will have no doubt that the coins were moved by you. Unless we are talking about a hacking incident. If you move the coins to a service to be mixed, they will lose the trace if you do it smartly. I am not sure how much into detail I am allowed to go with the new rule change, but let's just say that no one is forcing you to withdraw the whole amount in one transaction or do it immediately after a finished mixing session. The coins can be withdrawn in chunks and days or weeks after being mixed.
Of course, with all the congestion that's been happening for months now, these anonymization steps can become costly. It's up to everyone to decide how much they are willing to pay for it.
