What is the Default Password of a Root User, when a Mac is Purchased Anew?

This is a ways out of my wheelhouse so I may well be wrong, but I don't believe there is one. See: How to enable the root user on your Mac or change your root password - Apple Support

I don't recommend disabling SIP as it opens up your macOS to becoming damaged. Last year a simple Google Chrome update broke some Macs which had SIP disabled and prevented them from booting. This could easily happen again from any app update.

https://arstechnica.com/information-technology/2019/09/no-it-wasnt-a-virus-it-was-chrome-that-stopped-macs-from-booting/

There should be a way to around the issue besides enabling the root user. You may want to look into "visudo" to change how you can run a command with other permissions or without requiring a password. This is dangerous in itself too since it is easy to mis-configure permissions or not see how you are opening the system to vulnerabilities. Another option may be making your script more like a system script or Launch Agent/Daemon.

You should continue looking for other workarounds. I believe setuid may be an option, but you must still be extremely careful how you use it. Even with the risks of using setuid improperly I think it would be less risky than leaving SIP disabled. You may want to ask for assistance on the Apple developer forums for the best option to accomplish your task in a safer way. I know enough that I don't know enough about the best way to perform these options (or the best way to perform these options safely).

There really should be no reason to enable the root user account. Just use "sudo" which is safer since you can keep track of what user did what things when they authenticate using "sudo" plus when you are prompted for a password it should make you stop & think for a second about what you are doing to help minimize doing something horrendous. When the root user is enabled there is no tracking who the root user is and it makes it easier for someone to compromise the system.