GS minigames depend on your FPS.
Let me say that again.
GS minigames DEPEND on your FPS.
You can uncap your framerate (always a good idea for this) and disable the automatic limiter when the client is inactive (and that’s pretty much required for background farming) but doing other things means that your system is also trying to render those things at the same time. That will slow down the rendering of FFXIV. If you’re trying to watch a movie or play another game, your GPU’s pulling double duty rendering both FFXIV and that other thing, and the framerate for both will be lower than doing just one. EVERYTHING requires your system to render what’s going on, even just reading in a web browser.
In short, doing anything while also background farming MGP means that your FFXIV framerate will be lower. Whether it’s by enough to matter depends on a number of factors, including your specs and what else you’re doing. This is unavoidable without using a second, fully separate machine for doing other tasks.