How can I use one machine to access keychains stored on extracted disk of different machine?
You can just copy your old keychain to your new keychain:


Open Keychain Access from the Other folder in Launchpad.
Choose Add Keychain from the File menu.
Go to your drive > Users > Username > Library > Keychains (/Volumes/_DRIVE_NAME_/Users/USERNAME/Library/Keychains/).
Select the file named: login.keychain


Now you have your old keychain in your new one.