When you swap your btc coins to coins from another chain, any link is immediately broken.
It's not exactly broken, connection stays on exchange you used that can always get leaked, and maybe even on-chain if you used some cross-chain exchanges.
that's true, but i think he was referring to the swaps @examplens did on splash.tf (assuming they keep zero logs).
Even if that exchange keeps logs, they won't immediately send all logs of every user to all exchanges in the world. That is just ridiculous...
When you convert eXch coins to USDT or whatever, you can safely send those USDT to any other CEX you want without any worries...