Yes, we can have both.
There's a natural tension between your need for privacy and the authorities' need to be able to track suspects. An engineer would consider this an occasion for optimization, that is, designing a solution that does as much as possible to meet two conflicting goals.
One of the biggest threats to privacy has nothing to do with the content of your messages and files but rather with your habits. While we need a reliable identity credential from which we can control the use of information about ourselves, just the act of using an online identity creates a trail of online footprints showing where you've been. That trail of online footprints is prized by marketers, governments, political parties and others who want to know, without your consent, where you've been, what you've been shopping for, what ideas you've been exposed to, what ideologies are represented by the sites you regularly visit. Whether the snoops are commercial, noncommercial or governmental, they all value your trail of footprints for the same reasons: to anticipate what you will do next, to manipulate your perceptions, and to attempt to influence your actions.
The RentalCredential breaks up your trail of footprints so that it conveys no meaningful information. As with a rental car, you use it for a limited period of time, then turn it in to be used by someone else. Back in the RentalCredential office there is a secured record of who had which credential at which time in case an audit is legally called for. As long as you're not doing things that would lead the authorities to seek a court order to "pierce you privacy veil" the you can leave the cookie clubs clueless about your whereabouts and your habits.