Can somebody tell, if the EFCS works from LiveView?
I don't know about the D7500 specifically, but in the D810 EFCS works in the following way in LV: you still have to have the custom function for EFCS set and the drive mode in M-UP mode. You press the shutter release twice, in and out of LV the process is the same. The first button press doesn't apparently do anything but they made it so that the usage is the same ... the second activates the electronic shutter and closes via mechanical shutter. Outside of LV, the first button press opens the mechanical shutter and the second the electronic shutter (where the exposure begins).
In the D850 this is different; you can use EFCS either in Q, Qc or M-UP modes, and in Q, Qc mode you only have to press once (viewfinder or LV mode), and in M-UP mode you have to press twice in viewfinder mode, and once in LV. So now it works more closely as you'd expect, there is only one button press in LV.
I use MC-30A remote cord in conjunction with LV when I need EFCS. In viewfinder photography, I sometimes use Qc to get the benefits of EFCS, e.g., when using the 300 PF hand held. If I don't have the MC-30A packed in the bag for some reason, and if I'm using a tripod, I turn on Qc and exposure delay mode and press the shutter button with my finger. However, Qc uses EFCS only with the D850 (of the Nikon DSLRs), as far as I know.
I don't know whether one or two button presses are required with M-UP in LV mode in the D7500. But for sure you can use EFCS in LV, but you might argue that it's not the most convenient if you don't like to use remote control.