First, how is your external drive formatted? To avoid damaging the Photos Library an external drive must be formatted in either APFS format or Mac OS Extended (Journaled) format. Additionally, the drive can not have had Time Machine on it since it was formatted. There have been so many problems with using incompatible drives that the newest macOSs won't even allow a Library on a non-Mac formatted drive to open, since there is a chance of damaging the Photos database. See this:
Move your Photos library to save space on your Mac - Apple Support
If this drive is in a an incompatible format, stop running Photos with it immediately! A Photos Library can sit on an incompatible drive, but running it may corrupt the database. (This paragraph was required by law…)
If the drive is properly formatted and connected with a cable (not networked or in cloud, etc) then it should work just fine on an iMac. (Spotlight won'd include pictures in the SSD Library. Some location stuff acts funny.) Macs work well with more than one Photos Library, in fact. As you say, only one Library can be the System Library-- the one that Safari and other apps connect to. And only the System Library can connect to iCloud.
iCloud doesn't "sync" a phone to a Mac. When connected to iCloud, a Mac will synchronize with iCloud Photos, and the iPhone will also synchronize with iCloud Photos, so all three Libraries will be identical. And, by the way, if you have 119,960 pictures, then there are way more than 119,960 files. Files and pictures aren't the same thing.
The best thing is probably to simply create a new, empty Library on the external drive, make it the System Library, and then wait for it to fill up. This will take awhile, but you will be sure that the files are clean and the database has no corruption. You can keep the copied Library as a backup.
As you suggested, you can also just make the copied Library the System Library, connect it to iCloud, and wait for things to settle. That, too, will take some time. Photos will start scanning, looking for duplicates between Libraries and so on.
If iCloud Photos finds pictures on the iPhone that aren't in iCloud or on the Mac, it will copy those and do everything it can to make all the Libraries identical. It will also try not to transfer pictures that will create duplicates, but that isn't perfect.
What do you think?
