While I was doing Explore just now, I had a Fetility Clinic on my island and, like any person who reads nothing, I didn't read it as I thought I was once again just donating my genes and nothing more.

Boy was I wrong, I saw the status message of my Cockatrice is "Awaiting an offspring" (or something to this effect) and I go looking to see if she's pregnant or not and, plot twist, she isn't but she can't breed either.

So, what's going on here? What am I waiting for exactly?

If your pet is female and you accept the fertilty shrine's caption, your female will become pregnant from a random male who has already visited the shrine and donated his DNA.