Well people who have really wanted anonymity have had it for some 30 years with the current internet. History is a good teacher. Basically the answer is YES.
Before the fall of the USSR the "West" could not have total surveillance because they condemned the USSR for attempting to do so. Now for nearly 30 years they have desired to have total surveillance and have progressed towards that bit by bit.
But during all that time and still is, there have always been simple ways to communicate anonymously if one really wanted to. So the claim by governments that anonymous communications which the world has mostly enjoyed for thousands of years is now something of danger is a reversal of what humans enjoyed since "forever". Remember that the internet has replaced a lot of communications that would have been done in person (privately a lot of times).
So no I do not accept that anonymous communications is anything that needs any sort of censorship. And that such anonymity is anything to fear.
The criminals with or without SAFE can communicate and do so now, why would things change.
Obviously some will use SAFE's anonymity for bad reasons, but they would do so even without SAFE.