How did the Library of Alexandria actually end?

Slowly, and mostly through neglect. There is no reliable account of a single destruction. Caesar's fire in 48 BC burned dockside warehouses and probably some books; Aurelian's campaign in the 270s damaged the district; the Serapeum was destroyed in 391; and the story of the Arab conquest burning the books to heat the baths appears six centuries after the fact, in a single source, and is generally rejected.

The likelier account is institutional. The Ptolemies funded it lavishly and later rulers did not; scholars were expelled in a political purge in 145 BC and dispersed to rival centres; Pergamon, Rhodes and eventually Rome competed for the same people; papyrus decays and requires continuous recopying by staff who must be paid. A library is a budget line, and when the budget stops the collection goes quietly.

The myth of a single catastrophic loss is durable because it offers a villain and a moment. The reality - that the greatest collection in the ancient world was lost through funding cuts, political interference and slow institutional decay over several centuries - is both less dramatic and considerably more alarming, since it is a failure mode still available.