AFAIK that's largely untrue in the general case:
- Canvas painting only appeared in the 15th century, and took a while to spread, paintings before the 17th century (16th in some locations) tend to be on wood panels e.g. raphael and da vinci painted almost exclusively on board.
- Canvas is almost always coated with gesso which at the time would use rabbitskin glue as binder, no flax there.
- While flaxseed was the most common drying oil (in europe), walnut, poppyseed, and safflower, were also in use. And additives were usually mixed in to manage the viscosity of the paint, so even in the "best" case it's not like the paint would be just pigments in linseed oil unless it's a very early oil painting, which wouldn't have been on canvas.
- As for varnishes, not only was flax not the only drying oil for oil varnishes, varnishes could also be "spirit" varnishes (with a resin dissolved into a solvent like alcohol or turpentine), waxes were also sometimes used as or in varnishes.