I still can´t see it on my desktop but time machine found the volume.
Somehow, I imagine a default setting was changed
Click anywhere on the open desktop so that the Finder menus are displayed at the top of the screen
Click Finder, then click Preferences, then click General
Under the heading of "Show these items on the desktop", make sure that there is a check mark next to "Connected Servers"
Is there a way to "auto mount"?
You can have the drive mount on start up as follows:
Click System Preferences (gear icon on the dock)
Click Users & Groups
Click the Login Items tab (next to Password tab)
Click the + (plus) bottom at the bottom of the list
Navigate using the Finder like interface to select and highlight the hard drive
Click "Add" at the lower right
Regarding Time Machine backups.....I've tried this maybe 6 times in the last two years as a test and they have not been reliable. They work OK for a few weeks...sometimes even a month or two before I have corruption issues.
Disk Utility will fix them a few times, but after a few, it cannot and I have to start over again by erasing the drive and setting up Time Machine again. Then the same sequence repeats. Simply stated, I would not trust this method to back up my important data.
Maybe you will have better luck.