macOS (Catalina 10.15.7) - Save dialog - no way to filter/search
I think the crux is searching for a folder vs searching for a file.
Search in open and save works as well as Finder for me - just a different interface and limited space, less affordances are present, but as a long time user of search, perhaps I just have gotten used to the limitations and don’t even see clearly how it’s failing you.
When I need more space to search, I open a second Finder window and use spotlight search (sidebar search, tag search, Smart Folders, complex searches) to my hearts content and then drag the file I find into the save box.
It really helps you can drag and drop folders and files and they instantly update the file open and file save dialogs.
Any time this doesn’t work, I find one of two things.

The app isn’t using a standard dialog. You need to patch that app or select a new app so things are easy again
Spotlight itself is failing - this is easier to detect in Finder for me, but you can also detect this in app file dialogs if you are keen and know the results should show and do not.

The key win for me is having second display and drag/drop of files from outside the save box. You also might love a tool like DefaultFolderX - I used it heavily years ago but have gotten used to working with the OS and not needing extensions.

https://www.stclairsoft.com/DefaultFolderX/

The author is a long time supporter of Apple users and has superb value / customer support in my experience. The tool does amazing things - I just work on so many macs, that in stalling a bunch of tools doesn’t work for me unless I can’t live without that tool. Don’t take my not using it to mean it’s not awesome - it is absolutely amazing and well worth a trial.