Is it a known problem that Time Machine does (sometimes) not backup Thunderbird profiles completely?
Besides the Time Machine configuration there is another way to exclude files and folders from being backed up with Time Machine: the com_apple_backup_excludeItem attribute.

To find all files or folders with this attribute enter in Terminal:

mdfind "com_apple_backup_excludeItem = com.apple.backupd"


To check if the attribute is set for a single file or folder use:

xattr -x /path/to/item


If you get a result similar to this one: com.apple.metadata:com_apple_backup_excludeItem the item is excluded from a Time Machine backup.

Or:

tmutil isexcluded /path/to/item


If you get the result -    /path/to/item the item is excluded or +     /path/to/item the item isn't excluded.

To remove this attribute from an item use:

tmutil removeexclusion /path/to/item