I haven't studied it, but I believe ZCash https://z.cash/ is not designed to eliminate that record (that's what a blockchain is right?), but to encrypt it and then use zero-knowledge proofs to show the validity of the transactions.
That seems like it might be the best of both worlds in some ways: laws can be enforced when it's genuinely important to do so (the record is there so presumably if you seize people's keys you can verify transactions), but not so cheaply as to entail practical mass surveillance and tyranny. In principle presumably a possible weakness is later large-scale decryption if the encryption gets broken -- but that's arguably much less useful to the would-be tyrants and surveillers.
That seems like it might be the best of both worlds in some ways: laws can be enforced when it's genuinely important to do so (the record is there so presumably if you seize people's keys you can verify transactions), but not so cheaply as to entail practical mass surveillance and tyranny. In principle presumably a possible weakness is later large-scale decryption if the encryption gets broken -- but that's arguably much less useful to the would-be tyrants and surveillers.