Short answer: install macOS from Apple, create your account, then put your files back on top.
That is the shape of every Mac recovery, whichever backup tool you use, because the system volume always comes from Apple.
Apple silicon: shut down, then hold the power button until "Loading startup options" appears. Choose Options, then Continue.
Intel: hold Command and R at the chime.
From Recovery you can use Disk Utility to erase the drive, and then Reinstall macOS, which downloads it from Apple. Expect the download to be the slowest part of the whole day.
Setup Assistant asks you to create a user. Use the same short account name you had before if you can remember it, because file ownership in the backup refers to it and matching saves you a permissions detour.
If you are restoring from Time Machine, this is where you say so. Migration Assistant appears during setup and offers to bring your files, applications and settings across in one step. That is the smoothest path available on a Mac and it is worth taking when it is an option.
From Time Machine, after setup: open Migration Assistant from Applications, Utilities, and point it at the backup disk. Same operation, run later.
From a folder-based backup: restore into a scratch folder first, look at what came back, then move it into place. Restoring directly over a home folder that already has files in it is how people end up unable to tell what came from where.
From a Copy Shelter backup: open the app, choose the destination, and restore. If the Mac
that made the backup is gone, csctl reads the repository on any Mac with nothing but the drive
and your recovery key. It is free, it needs no installation, and it does not care whether you have a license.
~/Library. A
backup that quietly skipped hidden files gives you your documents and none of your configuration.Yes. Migration Assistant and any folder-based restore both work onto a new machine. Applications built only for Intel may need Rosetta on Apple silicon, which macOS offers to install the first time one runs.
From Time Machine, enter it from the menu bar and browse back through time. From Copy Shelter, browse the snapshot and restore the one file. From a plain rsync mirror, only if the file still exists in the mirror, which it will not if the mirror has run since you lost it.
Usually an hour or two, and mostly waiting: the macOS download, then copying your data at whatever your disk or network manages. Plan on an afternoon.
For the macOS install, yes: Recovery downloads it from Apple. For the restore itself, only if your backup is in the cloud. A Copy Shelter backup on a local drive restores with no connection at all, because licenses are verified offline.
Your passphrase opens the same backup, so the key is what you reach for when the passphrase has gone. If both are gone the backup cannot be opened, by you or by anyone, and that is true of every properly encrypted backup. It is the one failure with no fix, which is why the app shows the key once, offers to save it to a file, and says where not to keep it.
Read this before the day you need it, and restore one real file into a scratch folder while nothing is wrong.