It turns out that spindles aren’t sharp.
A spindle’s just a dull wooden rod that you twirl with your fingers as you spin. Back in the day, I’m conjecturing, it was a funny joke to compare the spindle a penis. That may be where the word “spinster” gets its start.
So when the audience of the day heard Sleeping Beauty, and heard about a spindle that drew blood, they were all like, oh I get it. Losing your virginity.