Rescue Eva Mac OS
If your Mac is using a firmware password, you're prompted to enter the password. Option-Command-P-R: Reset NVRAM or PRAM. If your Mac is using a firmware password, it ignores this key combination or starts up from macOS Recovery. Shift (⇧): Start up in safe mode. Disabled when using a firmware password. D: Start up to the Apple Diagnostics.
To restore a full installation from TM, you need an installer disc. It's possible to use a USB, but you need to install the OS on that from the disc in the first place (not necesarily on the same computer).

- Command (⌘)-R: Start up from the built-in macOS Recovery system. Or use Option-Command-R or Shift-Option-Command-R to start up from macOS Recovery over the Internet. MacOS Recovery installs different versions of macOS, depending on the key combination you use while starting up.
- The 6.0.4 version of Data Rescue for Mac is available as a free download on our software library. This app's bundle is identified as com.prosofteng.DataRescue3. The software lies within System Tools, more precisely Backup & Restore. The software is sometimes referred to as 'Data Rescue 3'.
The USB would need to be a minimum of 8GB; the installation DVD is Dual Layer.
Your understanding of Time Machine seems to be greatly lacking.
View Pondid's excellent articles, especially on how TM works.
TM is not bootable, full stop.

Firewire Target Disc mode; no other firewire devices may be connected to either computer during this procedure. You can't run a TM backup remotely anyway, it has to be from a booted volume.
UNIX forum; https://discussions.apple.com/community/mac_os/mac_os_x_technologies
Feb 7, 2013 3:26 AM
Start up from macOS Recovery
Determine whether you're using a Mac with Apple silicon, then follow the appropriate steps:
Apple silicon
Turn on your Mac and continue to press and hold the power button until you see the startup options window. Click the gear icon labeled Options, then click Continue.
Intel processor
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Make sure that your Mac has a connection to the internet. Then turn on your Mac and immediately press and hold Command (⌘)-R until you see an Apple logo or other image.
If you're asked to select a user you know the password for, select the user, click Next, then enter their administrator password.
Reinstall macOS
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Select Reinstall macOS from the utilities window in macOS Recovery, then click Continue and follow the onscreen instructions.
Rescue Eva Mac Os Catalina
Follow these guidelines during installation:
- If the installer asks to unlock your disk, enter the password you use to log in to your Mac.
- If the installer doesn't see your disk, or it says that it can't install on your computer or volume, you might need to erase your disk first.
- If the installer offers you the choice between installing on Macintosh HD or Macintosh HD - Data, choose Macintosh HD.
- Allow installation to complete without putting your Mac to sleep or closing its lid. Your Mac might restart and show a progress bar several times, and the screen might be empty for minutes at a time.
After installation is complete, your Mac might restart to a setup assistant. If you're selling, trading in, or giving away your Mac, press Command-Q to quit the assistant without completing setup. Then click Shut Down. When the new owner starts up the Mac, they can use their own information to complete setup.
Other macOS installation options
When you install macOS from Recovery, you get the current version of the most recently installed macOS, with some exceptions:
- On an Intel-based Mac: If you use Shift-Option-Command-R during startup, you're offered the macOS that came with your Mac, or the closest version still available. If you use Option-Command-R during startup, in most cases you're offered the latest macOS that is compatible with your Mac. Otherwise you're offered the macOS that came with your Mac, or the closest version still available.
- If the Mac logic board was just replaced, you may be offered only the latest macOS that is compatible with your Mac. If you just erased your entire startup disk, you may be offered only the macOS that came with your Mac, or the closest version still available.
You can also use these methods to install macOS, if the macOS is compatible with your Mac:
- Use the App Store to download and install the latest macOS.
- Use the App Store or a web browser to download and install an earlier macOS.
- Use a USB flash drive or other secondary volume to create a bootable installer.