Problems rescuing an old drive

Within Disk Utility you may need to click "View" and select "Show All Devices" before the physical drives appear on the left pane of Disk Utility.

If you don't see the physical drive at all, then you have a hardware issue of some sort.

If you cannot see or mount the volume on that drive, then that could be either a file system issue, hardware issue, or an inability of macOS to read an older drive using the HFS file system (notice no "+"). macOS 10.14 Mojave, or possibly macOS 10.15 Catalina was the last version of macOS that was able to read the older HFS file system.

The picture of Disk Utility shows the volume could not be unmounted. While it is not uncommon for macOS to have trouble releasing a mounted volume, I would be worried that the physical drive has a hardware issue which is causing it to be very slow where it is taking macOS a long time to actually mount that volume.

I would suggest running DriveDx (free trial period) and posting the complete report here using the "Additional Text" icon which looks like a piece of paper on the forum editing toolbar. You will need to install a special USB driver to attempt to access the external USB drive's health information. Of course, it could still be a cable issue or file system issue, but in my own personal experience such an older Hard Drive is very likely to have hardware issues especially if it has been sitting unused for years.....sometimes they just cannot come online properly. At least checking the health of the drive is relatively easy & costs nothing but a bit of time and can provide us with some important details.

I bought a cable that was supposed to be compatible (with power)....

Unless the drive is totally dead, the cable is my prime suspect.

Can you hear the drive spin up?

Have you cleaned the contacts both on the drive and the adaptor cable?

Have you tried turning the drive it on its side? The drive would have been mounted on its side in the old iMac. Older drives sometimes have bearing issues from running in a non-horizontal orientation for a long time. That may make them position sensitive.

Sometimes a drive dock works better than those adapter cable kits.

I agree with Allan Jones assessment.

Full sized Rotating Magnetic drives from desktop computers require a LOT of power to spin the platters. Your adapter-with-power may not be up to the job. A bona fide enclosure with a heftier power supply might do better.

Also, consider that problems internally on the drive might make it not respond.

Edit...

"Hi. I have managed to have disk utility recognize it as an external drive. I also get a pop-up saying MacOS cannot repair the disk iMac HD but I can access the files but not save changes. It recommends backing up the disk and reformatting. Finder now recognizes the iMac volume.[...]"

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Getting this to Read:

Using a toaster, plug it into the wall. I would also recommend verifying that it is compatible with your current operating system. You might need to download a few drivers, just for the toaster. From Amazon: hard drive toaster. Dive from there for compatibility and such.

Single-Out this User:

For Troubleshooting Purposes, create a temporary administrator user, and see if this occurs. If not, then it is an issue with your current user. If so, then it is an issue with the entire macOS. IMPORTANT: Create a Time Machine Backup prior to creating this user. That way, you can have something to restore it from, should anything go working after the creation(such as deleting your current one).