i don't have that monitor to say exactly how, but the idea of setting up the monitor goes like this: use calibration software to build a corrected color space profile specific to your monitor, such as a rec 709 profile, so it can correctly display that color space. then that profile can either be loaded/saved directly into the monitor, or if the monitor doesn't support that, then save the calibrated profile as a windows color management monitor profile.
i'm not sure what that website rec709 profile is for, but it sounds like the wrong thing. either the eizo color navigator (or whatever calibration software they have) or perhaps sypder 5 pro software from their website will have a rec709 display format choice. if your eizo monitor has an option in its menu, or a button, to switch color space modes, like between srg or rec709 etc, then you might just need to be on the rec709 mode and have it calibrate itself. using a pre-made rec709 profile alone won't correct any color shifts or brightness levels specific to your monitor. srgb is very close to rec709, but i think it defaults to 2.2 gamma.
supposedly adobe software requires a quadro card to send 10-bit to the display monitor, or a video i/o card. blackmagic has some starting for around $200, and there are some quadro cards that start around $200 too. the display i/o cards do not function as a gpu, so they don't use windows monitor profiles. so the monitor would have to support a calibration profile to handle adjustments, but this also removes the chance of windows interfering with the monitor display via a windows monitor profile. a quadro card would allow for a windows monitor profile to be used, but can be a bit tricky to get the quadro + geforce drivers installed for both cards. you also have to setup the nvidia control panel to tell premiere to only use the gtx 1060 for cuda, so the quadro won't slow it down and will only be for display.
i'm not sure about the best/correct gamma, it seems like its not a locked down spec of rec709 and changes depending on the monitor and mfg. i've seen 2.2-2.6, so perhaps 2.4 is a good middle ground. you also need to have the lighting in your room match the destination lighting, such as a dark room for a theater, or several lights for average home tv viewing.