Use of third-party apps to back up when Mac is not responding

The First Time Machine Backup must be done from within a Functional Operating Computer. If the computer is not starting up normally - suggest attempting to satrap in Safe Mode. Shutdown computer and disconnect all external drives. Restart and Immediately hold the - Shift - key. This will attempt to do a Repair Disk, clear cache files and only load Apple Software and Fonts. It will load slowly - Normal. If successful - it will look and perform differently - also normal. Then you can perform the Time Machine Backup. Again, suggest having a reliable External USB Drive at least 1 1/2 times as large as the size of the Internal Drive. Example - if Internal Drive ie 256 GB in size the External Drive should be 1.5 times x 256 GB = 384 GB in size.

There is not need to use Third Party Application to make Backups. The builtin Time Machine Backup application is System Preferences is all that is required. Back up your Mac with Time Machine