Why my Time Machine disk is not mountable suddenly

It is also possible you have a hardware issue of some sort. The backup drive could be failing, or you have a bad cable, adapter, or USB port. Also, make sure to disconnect all other external devices from the computer in case one of them is causing a problem.

You can also try booting into Safe Mode to see if that makes any difference....if so, then you most likely have some third party software installed which is interfering with the normal operation of macOS.

To check the health of the external drive you can try running DriveDx (free trial period) and posting the complete text report here using the "Additional Text" icon which looks like a piece of paper on the forum editing toolbar. You will need to install a special USB driver in order to attempt to access the external drive's health information, but even with this special driver some external enclosures will not allow the necessary communication to access the external drive's health information.

@etresoft is correct that the macOS system logs are nearly worthless for troubleshooting these days. It is very rare you will discover anything, but it is almost guaranteed the logs will drive you crazy & insane.

There is nothing of value in the logs. Erase your Time Machine drive and start it over again.

You access Logs via the Console. There is no guarantee that these apps write logs anymore, and even less certainty that they will contain useful information for you. These things are typically developers writing to themselves.

You can go to terminal and copy and paste this line. Copy the whole line as it extends across the visible window:

printf '\e[3J' && log show --predicate 'subsystem == "com.apple.TimeMachine"' --info --last 6h | grep -F 'eMac' | grep -Fv 'etat' | awk -F']' '{print substr($0,1,19), $NF}' 

There will be some information there, but not sure if it is what you are looking for.