I may have stumbled onto a fix for the "the library is unavailable" problem, though I will not consider it a fix until someone else uses this suggestion with success.
I kept thinking about the idea that a library could be considered unavailable. I checked that the drive with the library was mounted and I ran disk repair with the thought that one of those two things might make the drive seem unavailable to the computer. The fact that I couldn't access the directory of the library through the iTunes Home Sharing option lead me to believe that something was preventing Finder from reading (or seeing) the drive/library directory. What could prevent Finder from reading the directory? The drive was working and was seemingly healthy as I could access all of my images in that Aperture library with no problem. Repairing permissions came to mind. But, Disk Utilities will repair permissions on the startup disk only. Then I remembered having had to correct permissions in the past at the drive level for a reason I cannot recall. My Aperture library is on an accessory drive in my Mac Pro. There are 2 libraries on that drive. I had been unable to share images from either of those libraries.
By highlighting the drive name in Finder and then either going to "Get Info" under "File" in the menu bar or right-clicking on the drive name then clicking on "Get Info" opens a window with a section at the bottom labeled "Sharing and Permissions". I found that only my Home folder name had "Read & Write" permissions for that drive. I decided to add(+) "Admin" and give it "Read & Write" permissions. I think I restarted Aperture and iTunes (I may have restarted my computer. I do not remember.)
Having done that when I went to iTunes, "Choose Photos to Share with Apple TV" was available and I could see the entire directory of my Aperture library and it allowed me to choose which album(s) I wanted to share.
As I asked earlier, will this fix work for anyone else?
I apologize for the length of this post, but felt I need to explain all this hoping to make my actions clear.
Jerry