"No Backups" message keeps popping up

that thing that is showing is unknown type suggesting it is damaged. But it knows the size, 4TB, so the Drive is likely fine, just the directory is damaged

on the tiny View menu, choose "show all devices"

Then the display will slightly expand to show physical devices, Container disks inside the devices, and Volumes.

choose each in turn, starting with physical devices, and click "FirstAid", and let it run.

It sound more and more like the disk developed issue with its data and can not be repaired.

can not be repaired means it can not be used in its current form.

--------

For most users, the way forward would be to Erase the drive, by selecting the Physical device and choosing ERASE, then specifying the defaults: GUID partition Map and APFS format, and giving it a new name that you will remember easily.

Then open Time Machine preferences and tell it to use this disk. In a few hours, you will have a backup again.

if a drive has become disconnected, it may be seen again after a Restart.

If that does not show it again, it may need a disk utility RepairDisk.

What model is the WD drive and how old is it..?

While WD still make good hard drives, their drive enclosures are not the most reliable.

If plugging the WD drive into a different port and trying a different USB cable doesn't fix..?

Then set that WD external drive aside and get a more reliable external drive like the OWC Mercury Elite Pro to use for Time Machine. https://eshop.macsales.com/shop/owc-mercury-elite-pro

what version macOS?

what is the state of your macOS? up and running? is it damaged? does it need to be restored?

is it just a few files, one large folder, or the entire computer you wanted restored ?

what is the result of the terminal command:

diskutil list

it might look a little like this:

dev/disk7 (synthesized):

   #:                       TYPE NAME                    SIZE       IDENTIFIER

   0:      APFS Container Scheme -                      +239.8 GB   disk7

                                 Physical Store disk6s2

   1:                APFS Volume ⁨X14-SSD240⁩              97.5 GB    disk7s1

   2:                APFS Volume ⁨Preboot⁩                 26.5 MB    disk7s2

   3:                APFS Volume ⁨Recovery⁩                507.6 MB   disk7s3

   4:                APFS Volume ⁨VM⁩                      20.5 KB    disk7s4

and what does Disk Utility Repair Disk say about it?

The notifications lag the actual errors, and fixing the problem does not remove all the outdated notifications.

Clear all the notifications first, then check the status again.

When you enable automatic backups, Time machine MAKES those backups, even when the external drive is not available. Due to the magic of Apple File System, only Pointers to the files are accumulated, because there is no need under APFS to duplicate the existing file on the SAME drive. These local 'sets of pointers' backups are called snapshots, and are also stored on your boot drive.

if you go looking for backup data before time machine has 'unloaded' all the snapshots into actual external drive backup sets, you will be looking through backups in those snapshot files stored on the boot drive. Very confusing.

Once Time machine has 'caught up', and as long as the external drive is still available, you will be looking at backups on the external drive ONLY.

select your time machine drive and choose

FirstAid or RepairDisk, whichever is offered in your version.

In the long run, you DO want to see your newly-erased external drive specified as the backup drive in time machine prefs.

if only the old version is there, delete the old one, and try to specify the new one. You may need to briefly disable "backup automatically" to make the change.

You complained about in accurate notifications.

Notifications are never rescinded, and old complaints need to be looked at and dismissed as no longer applicable.

Your newly erased drive need to be install as a Time machine backup drive. And to get there, all old backup drives that are no longer available need to be removed.

The logjam of snapshot back-ups will clear up by itself, but only AFTER your newly erased drive is shown with the Time machine Icon as a Time Machine backup destination.