Mount Button Grayed out on Disk Utility when I try to connect an External Hard Drive

Maybe it's already mounted?

Does the external drive show up in Finder window sidebar?

Check Finder Preferences > General tab and see if there is a check-mark in the box for "External disks" under "Show these items on the desktop". Check also for a similar checkbox under the Sidebar tab.

Check your drive connection method. You're using a usb-a/usb-c adapter of some sort, are you confident in the quality of that?

How about the quality of the drive's external enclosure?

Also suggest you run Disk Utility > First Aid on the drive.

Please see > How to repair a Mac disk with Disk Utility - Apple Support

Another great diagnostic tool is DriveDX from Binary Fruit.

You might consider using the iMac to mount the drive, copy the contents to another external, and then erase and reformat the problem device and try again.

The adapter, drive dock, or enclosure you are using with this old drive may not be compatible with an M1 Mac. I tend to purchase items which utilize the ASMedia USB chipset and so far I have not had any issues connecting external drives to my first gen M1 Mac. The version of a USB chipset can play an important role in compatibility as well since one user mentioned they had two Samsung external SSDs, but one would work fine on an M1 Mac, but the other one would not as the two drives utilized different versions of the USB chipset.

What file system is on the old drive?

Is the drive being connected directly to the M1 Mac? Perhaps the M1 Mac isn't able to provide enough power for the drive, so you may need to use an external power supply for the drive or a powered USB3 hub even if the old iMac drive is a 2.5" hard drive.