Sending your BTC to a custody wallet doesn’t really have anything to do with avoiding KYC. It all depends on the source. Most people buy coins from CEX exchanges using KYC, then move them to self-custody wallets because they don’t care . When the time comes to shave off some or sell (through CEX exchanges), it’s much easier because the coins originally came from a KYC-verified source.

Unless you have a specific reason for the coins (maybe for privacy), or you’re trying to stay completely off the grid, you’d need a different source. But even then, it’s hard to know where your sats are really coming from, so it gets a bit more complicated than just being a regular DCA holder who doesn’t  care but yeah! you have the privacy that’s not linked to any KYC -source .

AFAIK, only the exchange has your personal details..unless there’s a data breach, or the government are on to it..you must be on their radar