Photo Library in iCloud?

You can sync it with iCloud but you will still have, at a minimum, optimized versions of the photos on your Mac or external drive.

iCloud Photos is a syncing service. Any photo you delete from a device where iCloud Photos is turned on will also be deleted from iCloud and from any other device connected to iCloud Photos. You can potentially reduce the storage space required on a device by turning on Optimize Storage. See the following Apple article for more information: Set up and use iCloud Photos - Apple Support

Apple does not currently offer cloud-only storage for photos. If that is what you want or need, you may want to explore 3rd party options such as Google Photos or Dropbox.

asgardfleet wrote: I store my Photos Library on an external SSD Drive, but it takes a very long time for Photos to load in.

You can run Photos from a library on an external drive. If you have a fast SSD, then it's totally transparent-- there is no loading time. You don't copy the library to your internal drive, you just plug in the drive and run Photos. In this case, the external drive isn't used for storage, really, it's just an extension of you computer.

The drive must be formatted in either APFS format or Mac OS Extended (Journaled) format. Additionally, the volume can not have had Time Machine on it since it was formatted. And the drive must be physically connected by cable-- not networked.

If you use iCloud Photos for syncing, then plugging and unplugging the library can be a bit more complicated as iCloud Photos can get confused. Even so, if you aren't connecting and disconnecting, even with iCloud Photos, running Photos from a fast external drive works well.

Check this:

Move your Photos library to save space on your Mac - Apple Support

If so, how do I move it to iCloud?

Photos needs a Photos Library to run, as FoxFifths explained, and this Photos Library needs to be stored locally on your Mac; storing it on an external drive with a wired connection, e.g. USB, is fine. But the library must not be stored on a network share- do not even think about putting the Photos Library on a NAS; and the library cannot be stored in a synced folder. If you move the library into a Dropbox folder or Google drive or iCloud Drive, it will be damaged by the syncing.

The older versions of MacOS let us drag a Photos Library or iPhoto Library onto iCloud Drive, and when we opened it there in Photos and worked with it, the internal links got broken and sooner or later the library could no longer be opened at all. Since macOS 12 Monterey any library we put on iCloud Drive will simply not be uploaded to iCloud and only stay local on our Mac. That is an improvement, but it is still not useful to put Photos Libraries on iCloud Drive.

For example - a Photos Library on my Desktop (syncing with iCloud Drive) is showing a crossed out cloud badge.

If you want the library in the Cloud then check out Adobe Lightroom CC. All your images are stored on their servers and accessed via the apps on any devices you have. Unlike Dropbox (which is basically storage only) or Google Photos (basic photo management) LRCC is a more powerful version fo Photos. There is a cost, of course, but it does what yo want.

https://www.adobe.com/ie/products/photoshop-lightroom.html