ianlever wrote: … Ultimately I am trying to achieve the position where the only photos which show in the opening screen are those which have not been moved to folders.
But, how would that work? I mean, if it worked like that, then when you put a picture in one album, then you wouldn't see it in the Library to add it to other albums. That would be quite the limitation!
In Photos, pictures aren't exactly "in" an album. When you "put a picture into an album," its name is added to a list of pictures from the Library that display together when the album is clicked, kind of like a music playlist, but for pictures. So two albums can both have the same picture name in their lists (like two playlists with the same song,) and that picture will show up when you click either album, but there's only one picture file-- it's just on multiple lists. And when you remove a picture from an album, its name remains in the lists of other albums, and the file still remains in your Library. Would you want a song in your Music Library to just disappear because you put it in a single playlist?
So albums give a specific view of your pictures. The picture of "Aunt Ethel at the Grand Canyon" can be in the "Aunt Ethel" album with other pictures of Aunt Ethel, and it can also be in the "Grand Canyon" album with pictures of other people at the Grand Canyon. And it can also be in a the "September 2015" album with other things that happened then. Each album is pointing to a single file stored in the Photos Library, so having pictures in multiple albums takes up no more storage space. As you can imagine, this is very powerful in organizing pictures.
By the way, pictures go in Albums, and Albums can be in Folders. If you have a Holiday 2024 folder, it can contain albums for Independence Day and New Years, so you can have much more organization. Folders can also contain other folders. I'm thinking that if you are "quite methodical at filing the photos" then this will be very useful for you.
I'm a bit confused by your description of the screen when you open Photos. Here is my screen:

The newest pictures are right in the middle, and the Albums and Folders are Quickly accessible. If it doesn't work like that for you, then maybe you need to use Customize & Reorder at the bottom of the window:

From there you can reorder the sections:

I have Albums right at the top so that section is right under the Library. If you don't like some sections, you can uncheck them so that they don't show. You can also, at Pinned Collections>Modify, put stuff you want in that collection so that you can find them easier.
Let us know if you have more questions about this stuff…