that after a certain amount of time the bitcoin becomes practically anonymous? As in, not being able to be traced back to the original KYC exchange, or in the case of "tainted" bitcoin, that?
Nope, It can be track if you even if you do this million times since every transaction is recorded on blockchain. Chain analysis is very powerful these days that can track a certain Bitcoin especially if the transfer has no discontinuity and just normal transfer of funds from different wallet.
Use mixer and your worries about anonymity will be solved since mixer use different wallet that is not connected to the source wallet of your funds. One of the best mixer to do this stuff is coinomize.biz because it has low mixing fee and guaranteed anonymity of mix since 2019.