I can't view Photos folders/albums on third-party sites, causing hours of additional work

Yep-- this has been a problem since macOS 13 Ventura, and no one knows why it doesn't get fixed. Until it does, I do one of three things:

1. I use the search field in Safari. I have attached keywords, titles, and captions to my pictures that have clues about the contents, so the Search Field may find them. Search also works for categories that Photos recognizes, like cat or church or river, etc, to help narrow down the hunt.
2. Like Yer_Man, I may prep the pictures for a search, like #1. I may select them all in Photos ahead of time and add a keyword like "TheseHere" so I can get them on a search. Yer_Man suggested using the Title field for this identifier. Unfortunately, neither the Keyword nor the Title show in iOS, so you may choose to put info in the Caption, as well.
3. I select the pictures in Photos and export them to a Finder folder. Then I point the "File Picker" in Safari to that album. 

You can see a bit more here:

Cannot browse Photos albums for website uploads on Mac - Apple Community

Also, I have left feedback about this for Apple, here:

Feedback - Photos - Apple

More comments there may help.

You and many others have been wondering why this change... but here's a workaround:

Select All the Images in your album for - say - the Farnagle property

Right click on one of the pics and choose 'Get Info'

At 'Add a Title' type in Farnagle - or whatever identifier you want for the property.

Then in the 'Add a Photo dialogue type 'Farnagle' in the Search box: Result is the contents of the Farnagle album.