Is the spindle of a spinning wheel really that sharp?

No, it must have been a splinter from the wooden spindle, although anything is possible in a fairytale.

Perrault's version of Sleeping Beauty first appeared in 1695 in a hand-written and illustrated manuscript called Tales of Mother Goose (French: Contes de ma mère l'oye) along with "Little Red Riding Hood", "Bluebeard", "Puss in Boots", and "Diamonds and Toads".

https://simple.wikipedia.org/wiki/Sleeping_Beauty