If you're on a KYC exchange then the chainalysis aspect is always there until you do something to break it. Your question was about other aspects and the key security is the other big one.

Anytime you share the bare pubkey fits the bill here so a spend to anywhere or even a signature proving you own said sats.

Depending on how big your utxos are you may want to consolidate when fees are low so maybe that'd be a good thing to do at the same time as switching to a new withdrawal address.