That's true, untill you decide to spend those unspent outputs. If you spend multiple unspent outputs funding several addresses, everybody will be able to link those addresses together. And with some clever analysis of the receiving address, the amount of outputs (change),... one could potentially start to trace you (like you said before, as soon as you filled in KYC info for a service and somebody is able to link you to this service, you are no longer anonymous).
A big exception to all this is the people that were ignorant enough to use a web wallet, exchange wallet, casino wallet. Next to making their clients vulnerable to loads of other attack vectors, the online wallet operator could link every address to a user.
There are even online tools that allows you to automatically trace all addresses that were used together in a same transaction, proving that the user hold all those private keys
Check this tool
https://www.walletexplorer.com/If you paste an address there, you will see all other connected addresses