If an X-T2 focuses but won’t actually fire, the main things to check are:

• SD card present and recognized; also try another card.
• Lens communication: if using a non-Fujifilm or adapted lens, enable “shoot without lens.” A faulty or unrecognized lens can cause similar behavior, so try another lens if possible.
• Shutter button/custom button settings, in case the button function was reassigned so a half/full press only activates focus.

Since you already tried cards, slots, reset, and modes, and there’s no error message, a hardware fault is also very possible. In the reported case here, the cause turned out to be a worn/broken shutter button that needed service by Fujifilm.

So after ruling out card, lens, and button-setting issues, the next step is camera repair/service.