That one Healing weave has different modifications and we know that Healing was one of the more common Talents during the Age of Legends, with a large range of strengths for the Talents' representation among its possessors. So their basic Healing weave covers so many bases, including the useful point that it seems most Aes Sedai can use it in some form, whereas other effects are limited by strength or other factors. They may also have preserved other weaves that they don't hold in the same veneration as Healing, such as one that makes the person feel refreshed even though they are deathly tired, and etc. It is also possible that they had something to do with Delving. Considering the religious trappings of the Aes Sedai as a whole, it's possible that it would be the Brown Ajah as a whole that was more interested in experimenting with the weave or preserving variant forms but were forbade to do so by Tower custom.

For example, there may have been a weave to repair a bad headcold, but since Healing covers that base, the more limited weave would have been one that a channeler in dire straits would have counted as less useful in an environment where there was no sophisticated technology to determine the exact ailment a patient was suffering from. Many of the Healing weaves that were known in the AoL were probably ultra specialized, and Healing could have been the last resort since it's possible for the subject to die.

I can imagine an Aes Sedai AoL doctor finding a wilder during the Breaking and because of a Foretelling, forethought or common sense, only teaching Aes Sedai Healing, and impressing upon her student(s) that it was vital that whatever happened, it was imperative that this one weave be taught to whatever surviving Aes Sedai or wilder she found. By the time the Aes Sedai gathered, a small community could have gathered for whom this weave became a badge of membership, an initiatory reward and these "primitive" survivors would become the precursors to the Yellow Ajah.