The name for our solar system is “Solar System”, after our sun, Sol.
What's the name for another stars system of orbiting debris?
Not my area but I think it might just be “system”.
They are generically called "planetary systems". If you're talking about a specific one then you use the star's name, such as "the Proxima Centauri system" or "the Solar system".
You mean the System of America?
It just strikes me as inconsistent, and I would imagine it's frustrating as a scientist when you need to reference it and distinguish between the concept/category vs "the place"
"The Solar system" is a specific and unambiguous instance of a planetary system. It's not a generic term for all planetary systems.