I share many of the concerns you've expressed, but worrying about the contract tracing API that was recently put forward is (hopefully) unwarranted. A faithful implementation of the joint spec published by Apple and Google would preserve your anonymity until and unless you published your secret keys. Moreover, the overall design is cryptographically sound (unless I missed something recently?) and appears to be decentralized (at least in principle) so in theory you won't have to trust the behavior of any binary blobs unless you choose to do so.
For more insight into the design (and security) of such systems, you might take a look at the TCN protocol (https://github.com/TCNCoalition/TCN) or the DP-3T protocol (https://github.com/DP-3T/documents).