No Photos library present

That is very odd. I can't imagine why the option key start is not working. What version of MacOS is on the Mini? - Or are you saying the library chooser comes up, but there is no library listed? If so, are you SURE the "use as system library" Button is greyed out in preferences. Where is the library stored on the Mini?

Can you check if your Mac Mini is starting in safe mode:

To check if you are in Safe Mode follow these steps:

1. Click on the Apple logo in the menu (top left).
2. Click in About This Mac.
3. Click on System Report.
4. Click on Software and check what the Boot Mode is listed as - it will say Safe if you are in Safe Mode, otherwise it will say Normal.

You need to set your library on the mac to be the system library in Photos preferences, and make sure the iCloud tab in photos preferences is also set as shown.

You also need to make sure your mac user account is signed into the same apple ID as your Phone.

When you do that, your library will sync with iCloud (both ways, so what is in your library now will also go to iCloud)

If you only want the photos that are in iCloud, create a new empty library (using the option key start) and set that one as the system library.

What does the sync status say at the bottom of the "all photos" view.

Has Photos created an "unable to upload to iCloud" folder.

If you create a smart folder with the condition "Photo - is - unable to upload to iCloud" does it show anything?

If you create a smart folder with the condition "Photo - is - referenced" does it show anything?

How are you connecting to the internet, and do you have any vpn or firewall/security software installed?

Which system versions are running on your Macs?

Are the system versions the same?

'Downloading 3,464 Originals Pause'

Your Mac is clearly trying to download files and can't. It might be a problem of the file format, the available storage, the library location, your network connection.

Where is your current Photos Library stored? In Photos on the Mac that cannot download check the location of the library. You can see it in "Photos > Preferences > General: Library Location" . Click the button "Show in Finder".

Is the library in your Pictures folder (the best place) or somewhere else? Do you have sufficient free storage for the file to download? There should be more than 20GB of unused storage.

And how is the Mac connected to the Internet? It should be a Wi-Fi connection, not a cellular hotspot.

Even for only 3,464 files it can take several days. Do not interrupt it.

If the file system versions are different on your devices, a Mac with a newer system version may not be able to handle files in iCloud uploaded by Macs with an older system version, particularly older videos.

Gav, to bring up zhe Library Choserr dialog, you have to hold down the options key ⌥ firmly and keep holding it down firmly, until the library chooser dialog appears. Only then release the ⌥ key.

If your main library is not shown in the dialog, it may be damaged or in an unsupported loacation.

The library should be in the pictures folder. Not in applications, and not on the desktop. Neither are supported locations for your photos library.

Try moving the library (NOT THE APP) into the pictures folder, again setting it as the system library, and seeing if it will sync then.

What about the other questions?