Lots of errors could mean there might still be a corruption lurking here after the recovery, or that the backup doesn't have the necessary access to the files and directories.
As for your question...
The system-wide /Library can be recovered from the macOS and software distributions, though not without some preparation and effort. And you'll want to preserve the specific installation kits your environment is built upon, as it's not always possible to re-download those kits.
The user-local ~/Library directory is part of your own local login environment. What's in this path isn't particularly recoverable from distribution. You'll have to rebuild what's in this area, if it's not backed up.
Use CCC or whatever to back up your most critical data—I'd "just" grab all of your login username directory under /Users, if you're suffering constraints around available remote storage or available network bandwidth—but for Time Machine to local, directly-attached storage or for Time Machine to local network-attached storage, or for similarly-local CCC processing—grab everything.
Grab the whole storage device.
The partial grab will be more work to piece together.
The Time Machine backup will be directly recoverable from the recovery or reinstallation sequence, too.
Why grab everything? Do you really want to be piecing together your environment from fragments, when you're seriously not happy, or seriously stressed out? That is, do you want to make this harder for yourself, atop the loss, theft, damage or failure of your existing macOS environment? Put another way, what's the price difference between a too-small direct-attached disk or a network-attached disk and a more-capacious storage device?
As an alternative to hosted backups... It's possible to rotate Time Machine backups with the addition of more target disks, as an alternative to hosted off-site backups.
