Adding music to iPhone that was previously deleted

This question has come up several times, but has yet to be answered sufficiently. I have an iPhone 12 Pro Max and I maintain my music manually through the Music app on my iMac (previously iTunes). There are times when I wish to delete a song or album from my phone and then re-add it - sometimes, the album comes across incorrectly as "Untitled Album" (another problem in itself). Other times, I want to delete an album so that I can re-add it with previously missing artwork.

When I delete the song from my phone either through Settings > General > iPhone Storage > Music, or through the Music app on my iPhone, the song seems to disappear. However, when I connect my phone to my iMac and open it in Finder, I can still find the supposedly-deleted songs via Manage Storage... > Songs. Only after deleting the songs through the Finder can I re-add them to my iPhone.

This is a huge inconvenience, as deleting an album means filtering through the songs via the Finder interface to find the many songs that appear on the album and deleting them one-by-one. In many cases, the song name may appear multiple times, either because it appears on other albums by that artist, or because other artists have a song of the same name. It is VERY difficult to determine which is the correct version of the song to be deleted.

Surely, there must be an easier way to delete a song from my iPhone once and for all.