Have you ever heard of a person bleeding to death from a papercut?

Even if spurs were sharp enough to cut into the flesh of the horse, and they usually weren't, the biggest health risk to the horse would be infection, not blood loss.

And I realize that while this response isn't up to the usual standards of this subreddit, this question is really a quite silly "common sense" question.

In the 19th C, most riders wore spurs and had for 2500 years at least. Cowboys wore spurs with rowels, a design meant to poke the horse but specifically avoid stabbing it in the side. The rowel would turn, letting the points move aside.

The dangerous spur was that used by some ladies riding sidesaddle. It was a simple long curved spike on the left heel, designed to be felt by the horse even if riding skirts and petticoats got in the way. A ride in the park could leave blood on the skirt if the rider used it too vigourously.

Still, no reports of equine exsanguination.