If the drive IS listed on Disk Utility, then it's possible to re-mount it.
"diskutil list" will show you all the drives currently mounted and unmounted.
What you're looking for is something like "/dev/disk1s2" (it might be another number)
Then the command diskutil mount /dev/disk1s2 will mount it again.
I can create a Workflow that checks what drives are unmounted and then present a list so you can select one to mount. Not hard at all. But as I was explaining in the last post, you're not going to be able to re-mount EJECTED drives, only UNMOUNTED drives.