Single Pilot Instructor reported an NMAC event during landing pattern training with another instruction flight on intersecting runway. Instructor Pilot departed the pattern after the event.

Narrative

Multiple airplanes including myself were doing practice take-offs and landings on Runway XX at ZZZ. One airplane; Aircraft Y was doing practice take-offs and landings on Runway XY at ZZZ. My student and I had to do a go-around to avoid a mid-air collision and ultimately resulted in us departing ZZZ. After the fact my student is terrified of non-towered airports and has no desire to go to them ever again. It is important to note that the traffic pattern of XX is Right and the traffic pattern of XY is left are in direct conflict with one another at multiple points.During the ordeal the only question that I asked on the CTAF was 'Any reason you can not use XX'? The other pilot responded 'Practicing cross wind landings'? I immediately looked at the windsock located to the right side of the departure end of Runway XX and it was limp.At one point Aircraft Y entered the left base for XY. Continued on what would appear to be an upwind leg of XX; made a 180 degree turn to enter the right downwind for XX which wound up being a right base for XY. Attempted to enter final for XY; then made a 360 degree turn; followed by another 180 degree turn to enter the left base for XY.Please understand; that after the recent mid-air collision at a nearby airport one would think that people would try to mitigate a mid-air collision and be more inclined to go with other traffic around them?