French have a word for people producing wine barrel material: "merrandier".

A "merrandier" will select trees (usually straight oak of a certain age), put them down and to size (the logs are then called "merrain", hence the name), split them using wedges, along the grain, them resplit using a froe (called "départoir", which is just a dull blade with a lever), then refine the smaller bit using a specific tool (called "queue d'hirondelle").

The resulting bits will be send to the barrel maker to be shaped into staves (called "douelles").

Riving is at the heart of the process, and is meant to provide longevity in resistance and impermeability.

To your point: a saw is really just a series of sharpened wedges

not exactly - saw makes the cut moving along the line of cut while axe - perpendicular. The significant point of difference here is that axe/wedge/etc. thus presses closed the vascular structure of the wood in the area of the cut, while sawing leaves it open - the result is that cut wood is easier to permeate by moisture.