Forgotten memories are ultimately lost for good
This FAQ expands on the content of "I would never send my kids to school" by Piotr Wozniak (2017)
We never forget?
Rapid progress of research
Today, we are rapidly developing tools that make it actually possible to associate memory with individual neurons and show that loss of synapses leads to permanent forgetting. Two-photon imaging, optogenetics, single-cell recordings, and other tools combine breakthrough techniques that get rapidly used in concert to elucidate the function of the brain at the neural level. We can see connections forming in a matter of hours. We can also see connections lost. Some researchers even use the term "synapse deleting". See: Optical erasure of synaptic memory traces.
However, the unshakable and more obvious evidence of the permanent loss of memories comes from computing sciences. Memories are primarily lost to interference. Once one memory is taken over by another memory, restoring the old memory is only possible be re-learning. Post-interference re-learning is, at least in part, based on laying down new memories.
For details see: Mechanism of forgetting
Role of forgetting
The first obvious role of forgetting is to prevent running out of memory storage. The number of memories we store in working memory cannot possibly enter long term storage of high stability as we would quickly run out of space even if individual memories would require a tiny extra protein buildup. However, even more important a role of forgetting is the change to the neural structure of memories. The wiring diagram of memories determines their quality and computational value. For this reason individual synapses, dendritic spines, neural processes and individual neurons may be subject to deleting, shedding, retraction, apoptosis, and the like. The whole system of transferring memories from short-term to long-term storage and building up stability is based on the computational golden mean in the plasticity-stability dilemma. Forgetting plays a key role in that system.
Myth of permastore
The myth of permanent memories comes from the fact that through a search procedure we can occasionally retrieve memories that have long been thought forgotten. Search is different from automatic retrieval. In automatic retrieval, a simple connection can be used to bring up a memory. In theory, a single synapse might suffice. In memory search, a neural procedure is involved. It has a form of memory scanning: probing activation for a possible retrieval. Memory search involves additional memory connections that must be involved to reactivate pathways needed to retrieve a memory. This scanning process is similar to creative thinking and is highly unpredictable. The less likely the retrieval, the bigger the effect, and the resulting awe makes people believe memories are actually never lost.
The second contributor to the myth is the concept of reconstructed memories. Some connectivity leftovers may be used to reconstruct seemingly lost memories (e.g. episodes from one's childhood). Reconstruction is an unreliable process and should not indicate memories survive actual forgetting.
Finally, cue-dependent forgetting can provide an impression that all we need for retrieval is to restore the original cues (see Endel Tulving's article in American Scientist 1974).
Misleading headlines
As for the article in Wired, the headline is very misleading. Proving that we can retrieve memories that were thought forgotten says little about the permanent forgetting. If you find a hopping frog in the forest, you do not conclude that no frog has ever died.
Incidentally, I agree with you that Dr Bjork's contribution to the science of memory is tremendous. All his views are based on solid science, experiment, and careful consideration.
For a comprehensive discussion see: Lost forever or temporarily misplaced? The long debate about the nature of memory impairment by another icon of memory research: Larry Squire.
Further reading