You can still see your buddy in the profile, as well as on the little 'trainer card' pop-up that everybody has. For example, yours is a Legendary Egg named 'Steve'. It doesn't show up on the sidebar on the index page, but it's definitely there.
Incidentally, although the Catcher is down, it's slowly being put back together, and right now posts will count to leveling up your buddy. I found this out on accident and was thus determined to break my drought of posts. Right now my buddy is at Level 22, and I'm not sure what level it's going to evolve at yet - hopefully something sooner than later.
Finished the Raid and I managed to evolve Spider Pig when all was done! And with perfect timing too. Now to work on this Level 24 Egg.
(40-27)*5 - 1 = 64 Okay, so my egg is Level 27, and needs only 64 posts left in order to hatch. ...wait. ... ... ...63.