Yeah, the precise connection between betula and bitumen does seem a little murky to me, but for all intents and purposes, “betula” means “birch.” Linnaeus simply adopted the common Latin word, I gather. The Online Etymology Dictionary says,

genus of the birches, from L. betula “birch,” from Gaul. betu- “bitumen” (cf. M.Ir. beithe “box tree,” Welsh bedwen “birch tree”). According to Pliny, so called because the Gauls extracted tar from birches. Birch tar is still sold as an analgesic and stimulant and made into birch beer by the Pennsylvania Dutch.

But sap is clear — nothing like tar! And birch beer production is hardly limited to PA Germans. So I can well believe that a web search led you in circles.