By the end of 18th century the theory of four elements or the mystic duality between planets and metals, which among other things informed the formation of alchemic symbols, did not have much currency with chemists. Neither did traditional alchemic objectives like the elixir of life everlasting, panacea (universal cure), or transmutation of metals. The old notation suggested connections that weren't there, and missed the ones that were. There was a similar question on Reddit, where a commenter put it bluntly:"Until after 1800, they had no clue of chemical structures nor a notation for writing them. Although there were lots of empirical observations, alchemy actually contributed very little to the modern science of chemistry as a theoretical structure." He sources Buckingham's Chasing the Molecule for the sentiment.
At the end of of 18th century chemists started figuring out real relations between different compounds, and anticipating molecular structure and valency bonds. In 1803 John Dalton explicitly introduced the idea of atoms "hooking up" together to form molecules, and in 1808 replaced most of alchemic notation with his own, incorporating the idea of elements forming compounds. Berzelius introduced more or less modern notation in 1813-14, flushing out what little was left of alchemic symbology.
Some of the symbols survive for incidental reasons. Diamond symbol is used by jewelers because it literally resembles a cut diamond. Caduceus (the rod of Hermes), appropriated by alchemists to represent amalgamation of sulphur and mercury, is used as a symbol of commerce, and in the US of medicine due to a historical comedy of errors. It is unknown if John Wallis had Ouroboros in mind when he picked $\infty$ for infinity. Few mathematicians know (or care) that $\triangle$ is fire, $\nabla$ is water, $\square$ is urine, $\ominus$ is salt, $\oplus$ is sal armoniac (a mineral form of ammonium chloride), and $\odot$ is gold. Dalton, interestingly enough, used $\oplus$ for sulphur, and $\odot$ for hydrogen, they are that flexible. It is likely that mathematicians made up those symbols independently rather than picked them up from alchemic tables. But here is one seen in some logos, which is definitely borrowed, the Squared Circle, the 17th century glyph for the Phiolosopher's Stone.