Archiving it and publishing it are different things.
More importantly, they may sabotage their mission: If Spotify shuts them down, their exiting archives and especially future archives may be effectively lost.
I guess I should say more accurately: Their mission is to both archive it and publish it. They seem to be explicitly against copyright, on principle. Which I greatly respect.
Fortunately, Spotify does not have that power. Annas Archive is not based in US or EU jurisdictions. They can make access for normal people a bit harder, but not shut it down.
They stated that they would pass the information on to other archivists and public/private trackers no? They obviously have backups, since there are multiple users seeding Gbs and even TBs of data. Mirrors can be created as well, like TPB.