Thank you jameson.h. Unfortunately, none of this worked. iCloud Music Library was already enabled, and none of the issues in the other link apply to me. This is what is happening:
I create a playlist. I browse for an album from a band I love. I find the album, click the 3 dots next to its name, and select Add to a Playlist...<name of playlist I've just created>. Check the playlist -- it's empty. I go back to Browse, click on the first song in the album I've tried to add, and choose to add that SONG to my playlist. It shows up. So from iTunes, I simply can NOT add entire albums, but CAN add songs to a playlist. This is unbelievably frustrating because a lot of my listening is album listening, and adding each song one by one takes a lot more time.
Weirder still is that I CAN add an album to a playlist on my phone. However, when I do, the album appears in the playlist out of track order. I'm not even sure what order it gets added in. It's not alphabetical, it's not by track number, it's not by length. It's seemingly random.
If this isn't resolvable on this discussion, is it a problem I could take into an Apple Store? I would really like this solved before my next Apple Music subscription payment. Thank you.