The problem with the statement "choose a different coordinator" is that there are to date, as far as I am aware, no other coordinators to choose from.
IIRC Chaincase wallet has it's own coordinator.
Coordinators cannot provide any data to surveillance parties, registration for each input happens over a separate Tor identity. A coordinator can only refuse to include an input, not identify it.
Except zkSNACKs are paying one or more blockchain analysis companies to investigate each input as much as possible and link that input to as much third party data as possible to then tell them whether or not they should censor it.
Additionally that means those company also know owner of that input have plan to use CoinJoin or have interest to protect their privacy.