Why, on Facebook, doesn’t the person I search all the time, who is not my friend, ever show up in my suggested friends?

Probably because there’s a chance that person doesn’t fit any of the information in your profile.

Facebook suggests friends to you based on your profile details.

Where do you attend school?

Your place of employment

Where do you reside?

People that your Facebook friends suggest as friends to you

As a result, Facebook suggests people for you to add to your friends list.