I use one of these for filling cooling systems....
https://www.matcotools.com/catalog/p...TEM-AIR-PURGE/
From what you are describing and show it is almost exactly as this works for cooling. It draws vacuum down and makes sure there are no leaks, then with the hose dropped into the coolant you open the valve and it forces it into the entire system. I haven't had to burp a system since 1996 when I bought this.
A system like this for brake fluid would be incredible!!!