From your description, I learn that your unmountable and corrupted USB drive can't be fixed by First Aid but it shows up in Disk Utility.
My question is, have you tried manually mounting the USB drive in Disk Utility? If the corrupted USB drive still can't be mounted by your hand, it indeed has serious problems.
It seems that you don't way to pay for restoring data on the corrupted Time Machine USB disk. To be honest, you are lucky that the unmountable drive shows in Disk Utility. It means that you can repair the corrupted USB drive on Mac with some tricks. Otherwise, you have to send the drive to a hardware repair lab which asks for much more money.
And although there are ways like running the FSCK command in Single User Mode and CHKDSK to repair the hard drive on Mac, they are usually invalid as First Aid can't help repair it. And running commands on a Mac is risky as for a freshman a mistake may make your Mac goes die.
To fix the corrupted external drive that can't be fixed by First Aid, the best reliable and secure way is to restore data on it with a professional data recovery tool and then reformat the USB on your Mac.
As you've learned, iBoysoft Data Recovery for Mac is the best choice per the price and high data recovery rate. I've used this data recovery tool a few months ago to get my files off my corrupted external hard drive.
This tool allows you to get the lost data off the unmountable USB with simple clicks:
- Free download and install the app on your Mac.
- Open it and select your unmountable USB on the main interface.
- Click Search for Lost Data to start scanning the files on the USB drive.
- Preview the files and select your wanted ones, then click Recover to get them back.