Don't worry. The good news is, the exFAT drive doesn't have physical problems because you can see it in Disk Utility, though greyed or Not Mounted.
Read this: What is exFAT and How it Performs on Mac
I have ever used iBoysoft Data Recovery for Mac to recover my lost data from a corrupted external hard drive that can't be mounted on my MacBook. You can free download it to rescue your data on the unmounted exFAT drive.
After getting all the files, just reformat the drive to make it mountable again on your Mac.
1. Disks formatted as exFAT on Windows are not compatible with OS X
Windows formats a drive to exFAT with a 1024k allocation unit size by default. But in OS X, the default allocation unit size for exFAT is 128k. So, if you've formatted the drive to exFAT on Windows but use it in OS X, it is no wonder that the drive won't be compatible with OS X. So, you find the exFAT drive not mounting on Mac.
In this case, save the files on the exFAT drive to your Windows computer, and then reformat the drive to exFAT on Mac to enable it to be compatible with OS X. Then, you can mount and access the drive on your Mac. After that, return to Windows to copy the files back to the exFAT drive.
2. The exFAT hard drive has logical errors
But on the latest macOS, the compatibility of the exFAT file system is greatly enhanced. If you can't mount the drive on Mac, there's also a possibility that the drive has logical errors. You can run First Aid in Disk Utility to check and repair the errors on it and then use the Mount button to manually mount it.
But First Aid can only fix tiny logical errors. If the external hard drive still not mounting on Mac, it may be seriously corrupted. In this situation, rescuing the data on the drive is the primary thing. There's no other feasible way to get files from an unmounted drive except using professional data recovery software.
Hope my tips can help you!