New Mac doesn't have enough hard drive space to restore my iPhotos from Time Machine

To recover your photos from an iCloud Photos Library create a new, empty Photos library, either on an external drive or in your Pictures folder. Any external drive used for Photos should be prepared as described here: Move your Photos library to save space on your Mac - Apple Support

• To create a new, empty Photos Library launch Photos while firmly holding down the options key ⌥ and double clicking the Photos icon. Release the options key ⌥ when you are seeing the dialog to select a library. Click the "new" button and select a folder on your external disk or your Pictures folder to host the library.
• When Photos opens on the new Library open the Settings and click the button "Use as System Photos Library " in the General tab of the Settings. Then click the iCloud tab and enable iCloud Photos. If the library is on your internal drive, click also "Optimize Mac Storage". This way you Photos will only download previews and see most of the originals in iCloud to save storage on your internal drive. But if your library is on the external drive use "Download originals" instead of "Optimize Storage". Then you do not need an internet connection to work with your photos and Time Machine can make backups of all your photos.

What was your long term plan when you purchased the Mac Studio with a much smaller internal drive? Were you planning to have your Photos library on an external drive? Reduce the number of photos and videos kept in your library? Destroy the rest or archive them? Keep all of them in the library at lower resolution? Something else?

Have you been using iCloud Photos? You might be able to leverage that to assist you with some of those options.

Do you still have the old Mac, or did you trade it in?

It's understandable that migrating a large photo library from your old iMac to a new Mac with a smaller SSD can pose a challenge. Here are a few options you can consider:

Selective Restoration:If you still have access to your old iMac, you might consider using your Time Machine backup to selectively restore only the most important or recent photos and videos. You can manually copy these files to an external drive or a cloud storage service and then import them into the new Mac's Photos app.

External Storage: If your new Mac has Thunderbolt or USB-C ports, you can connect an external SSD or hard drive to store your photo library. You can then point the Photos app to this external drive as the location for your photo library. This allows you to access your photos and videos without consuming internal SSD space.

iCloud Photos: Consider using iCloud Photos to store your library in the cloud. This way, your entire library doesn't need to be stored locally. You can set up iCloud Photos on both your old iMac and new Mac, and access your photos and videos from any device while keeping the full library in the cloud.

Network-Attached Storage (NAS): If you have a network-attached storage device with sufficient space, you can use it to host your photo library. This way, you can access your photos over the network without taking up space on your Mac's internal drive.

External Storage for Archives: You can create archive folders on an external drive, organize your photos into these folders, and then store them on the external drive. While they might not be as seamlessly integrated as using the Photos app, you can still access your archived photos when needed.

Clean Up and Optimize: Consider going through your photo library to remove duplicates, low-quality images, or videos you no longer need. This can help you free up some space and make the migration process more manageable.

Upgrade Internal Storage: If your budget allows, you might consider upgrading the internal storage of your new Mac to a larger SSD. This would give you more space to store your entire photo library without relying on external solutions.

When deciding on the best approach, consider your storage needs, budget, and how you want to access and manage your photos. Each option has its own advantages and limitations, so choose the one that aligns best with your preferences and requirements.

Selective Restoration:If you still have access to your old iMac, you might consider using your Time Machine backup to selectively restore only the most important or recent photos and videos. You can manually copy these files to an external drive or a cloud storage service and then import them into the new Mac's Photos app.

Do this and you lose your edit history, and need to choose between the originator edited versions. There is a serious risk of dataloss here.

External Storage: If your new Mac has Thunderbolt or USB-C ports, you can connect an external SSD or hard drive to store your photo library. You can then point the Photos app to this external drive as the location for your photo library. This allows you to access your photos and videos without consuming internal SSD space.

Every new Mac has these ports. But you (or ChatGPT XYZ or whatever it is) neglect to point out the most obvious: the extant drive needs to be formatted Mac OS X Extended, Journaled or apfs.

iCloud Photos: Consider using iCloud Photos to store your library in the cloud. This way, your entire library doesn't need to be stored locally. You can set up iCloud Photos on both your old iMac and new Mac, and access your photos and videos from any device while keeping the full library in the cloud.

The problem with this is that you can't do that without first restoring your Library from Time Machine, and you still don't have enough space for that.

Network-Attached Storage (NAS): If you have a network-attached storage device with sufficient space, you can use it to host your photo library. This way, you can access your photos over the network without taking up space on your Mac's internal drive.

You cannot run an iPhoto or Photos library from a NAS. You risk silent corruption of the Library and Dataloss.

External Storage for Archives: You can create archive folders on an external drive, organize your photos into these folders, and then store them on the external drive. While they might not be as seamlessly integrated as using the Photos app, you can still access your archived photos when needed.

Might not? Will not. Also, again as with selective restoration, you're forced to dump one version of another of the photographs, hence dataloss.

Clean Up and Optimize: Consider going through your photo library to remove duplicates, low-quality images, or videos you no longer need. This can help you free up some space and make the migration process more manageable.

And again, how do you do this without first restoring the Library, which you have no space to do?

The option is to format an external disk to Mac OS X Extended, Journaled or apfs. Restore the library to there, and then decide what you want to do.

When deciding on the best approach, consider your storage needs, budget, and how you want to access and manage your photos. Each option has its own advantages and limitations, so choose the one that aligns best with your preferences and requirements.

When offering advice that risks dataloss you need to warn people.