Apple

Yes, what you want is 2 clicks away (or taps).

When you select a song I. Your library, you are given the option to see it in Apple Music. Select Go to Album. It will take you to the album, but only list the songs you have added.

From there, there’s a link at the bottom of the song list Show Complete Album. Select that and you’ll be taken to the full album. Hit the giant play button at the top and it’ll start from track 1 and right through.

You can also search for the artist/album manually and simply play the entire album in the same way.

The way I was doing it is to disable the setting that automatically ads songs to your library. That way I added songs I like to playlists and only added full albums to the library. But now they just made a change that replaces the “love” button with a “favorite” button which will add that song to your library automatically. Basically it’s gonna be a pain in the ass to manage albums in Apple Music in any meaningful way. I hope they fix this somehow.

I don’t believe there’s a way to do that. Making a playlist and putting it in album view does seem like the obvious way.

Are you sure you are not looking at downloaded albums? Can you give an example of an album with a missing song?

When you add an individual song or a playlist to your library it will not add the whole album to your library.

If you’d like to add an entire album to your library then you have to manually add the album to your library.

The quickest way to do that if you’ve got a song selected is to select the song in the player view, open up the menu and go to album. Once on the album page you can add it to your library.

Or if you search for the song in the search page, you can long press an individual song or open the submenu and go to album where you can then add the whole album to your library.

Perhaps you could keep a separate library for your complete albums?

I used to do something similar when I wanted to separate my lossless music from the rest.

https://support.apple.com/en-au/guide/music/mus7663a920a/mac

Thinking through with Shortcuts, I wonder if you could create a dictionary that has the highest track number for each album, and then iterate through again and see how many songs you have for each album, and see if they match