How do I keep photos organized on iPhone 14 Pro library?

What you described is how you organize photos. You move photos into folders. They aren't duplicated at all. They are just pointed to the folder, yet remain in the recents library. Folders don't take up any extra memory, they just let you store photos more logically in folders you can name as you choose. It's up to you to organize them as you want.

See how to do that here, if you're not familiar with this --> Edit and organize albums on iPhone - Apple Support

I don't want the library and then create albums but still have the library with 1,000s of photos unorganized.

Then use a different application that works the way you want.

Photos, (like iPhoto, Aperture, Music, iTunes) is a database. This means it can be very flexible and powerful, if sometimes a little confusing. The library is the core of the app. Every image is in the library. Albums are virtual, and are just subsets of the library. In fact, Albums are not places, they are lists - instructions to the app to 'display this group of images together'. So images do not get "moved" to Albums, they get added to that particular list. This means that an image can be in more than one album but use no extra disk space. Similarly in the Music app, all the tracks are in the Library and can be added to any number of playlists.

In a Mac:

If you're trying to organise your Library and want to find images not in albums, then a simple Smart Album

File -> New Smart Album

Album -> Is Not -> Any

will find these images. As you add them to albums they will disappear from the Smart Album.

On iPad or iPhone, which do not currently support Smart Albums:

Check out HashPhotos

https://apps.apple.com/ie/app/hashphotos/id685784609

It integrates with Photos on your iOS device and adds a Smart Album feature. Using this app a search

!#inalbum

Will find images not in an album

(Hat tip to User ukSamo for the recommending this app.)