If you display Time Machine's log messages, maybe you can figure out what TM is doing when it gets stuck (or maybe not!).
If you're comfortable using Terminal.app, you can show TM log messages by entering the following 'log' command:
log stream --style syslog --predicate 'senderImagePath contains[cd] "TimeMachine"' --info
Then, manually start a TM backup. Log messages will display in the terminal. Hit Ctrl-C in terminal to quit the 'log' command.
So I ran it and this is the error that came up. Obviously it says something is too large but I'm not a great log reader so not sure how to resolve as this is the folder that I'm not familiar with.
2022-06-02 01:14:14.580791-0500 localhost backupd[1074]: (TimeMachine) [com.apple.TimeMachine
iskImages] Creating a sparsebundle using Case-sensitive APFS filesystem
2022-06-02 01:16:11.956387-0500 localhost lsd[533]: (TimeMachine) [com.apple.TimeMachine:General] Failed to enumerate URLs under /private/var/folders/vf/lp9klqqn5nj31qczgv9bdwyw0000gn/X/1A0E7D74-9912-594C-9FF3-B4FEF649DC0E for SnapshotStorage reuse with error Error Domain=NSCocoaErrorDomain Code=256 "The file “1A0E7D74-9912-594C-9FF3-B4FEF649DC0E” couldn’t be opened." UserInfo={NSURL=file:///private/var/folders/vf/lp9klqqn5nj31qczgv9bdwyw0000gn/X/1A0E7D74-9912-594C-9FF3-B4FEF649DC0E, NSFilePath=/private/var/folders/vf/lp9klqqn5nj31qczgv9bdwyw0000gn/X/1A0E7D74-9912-594C-9FF3-B4FEF649DC0E, NSUnderlyingError=0x128f3a010 {Error Domain=NSPOSIXErrorDomain Code=34 "Result too large"}}
2022-06-02 01:16:14.087234-0500 localhost lsd[430]: (TimeMachine) [com.apple.TimeMachine:General] Failed to enumerate URLs under /private/var/folders/vf/lp9klqqn5nj31qczgv9bdwyw0000gn/X/1A0E7D74-9912-594C-9FF3-B4FEF649DC0E for SnapshotStorage reuse with error Error Domain=NSCocoaErrorDomain Code=256 "The file “1A0E7D74-9912-594C-9FF3-B4FEF649DC0E” couldn’t be opened." UserInfo={NSURL=file:///private/var/folders/vf/lp9klqqn5nj31qczgv9bdwyw0000gn/X/1A0E7D74-9912-594C-9FF3-B4FEF649DC0E, NSFilePath=/private/var/folders/vf/lp9klqqn5nj31qczgv9bdwyw0000gn/X/1A0E7D74-9912-594C-9FF3-B4FEF649DC0E, NSUnderlyingError=0x14a729e60 {Error Domain=NSPOSIXErrorDomain Code=34 "Result too large"}}
2022-06-02 01:20:37.336679-0500 localhost backupd[1074]: (TimeMachine) [com.apple.TimeMachine
iskImages] Failed to create '/Volumes/.timemachine/DiskStation918._smb._tcp.local./9C24A032-0B70-49AF-A5F3-EA1D988A4AFD/macbooktm/0C7120ED-DEFF-5FF1-B3C0-227A563B69B8.sparsebundle', results: {
}, error: 60 Operation timed out
2022-06-02 01:20:38.160018-0500 localhost backupd[1074]: (TimeMachine) [com.apple.TimeMachine:General] Mountpoint '/Volumes/.timemachine/DiskStation918._smb._tcp.local./9C24A032-0B70-49AF-A5F3-EA1D988A4AFD/macbooktm' is still valid