You misunderstand, completely. A device needn't download all activity, or indeed download anything additional at all. It knows which ads it clicked on and it knows the conversion signal for each ad the user saw. It can thus easily count the user's conversions on the device, then privately aggregate the result with its peers using secure multi-party computation (or a secure enclave).

Even the ads can be anonymously downloaded using a shuffler/mixer, such that nobody knows which ads out of the universe of ads the device chose to target the user with.