Paper will be destroyed, eventually, in the presence of incorrect humidity levels. It’s just a matter of when.
Yes, it isn't zero-effort, you can't just chuck your papers down a mineshaft and come back for them 50 years later, but it isn't in any way an insurmountable problem.
Fire, one of the biggest concerns in archival practice, is also much harder to manage in underground facilities. Document archive facilities are designed as distinct fire cells to mitigate the risk, and incorporating these features into bunkers would likely require rebuild from scratch. Refrigerant fire suppression systems are difficult to install because of inadequate evacuation routes; Cold War era underground facilities rarely have more than one staircase and one ladder (and the ladder shaft might be full of sand!).
The disused military bunkers of the United States are littered with failed document storage companies that have been unable to sufficiently retrofit them for this use. Seriously, I would guesstimate as many as one in six former ICBM silos was, at some point, owned by one of these operations. More than one per field usually. Almost none have succeeded, and the exceptions have the appearance that they are more hobby projects than viable services. In most cases they didn't even finish stripping the old equipment, which is a very costly process since it has to be cut up underground to hoist out.
Perhaps records only need to last lifetime + 70. That should make storage concerns a lot easier.