General aviation instructor reported a near miss with another aircraft in the traffic pattern at a non-towered airport. The instructor maneuvered away from the other aircraft to avoid the conflict.

Narrative

My student and I were doing pattern work at SGU. While mid field on a left downwind with the intention to land runway 19 another airplane did a mid-field cross over from the West at pattern altitude. The other plane did radio calls indicating what their intentions were upon entering the airport traffic area but; the altitude and distance information was terse at best. Apparent from their radio calls they were transient to SGU. We had reported that we were established on the left downwind mid field. There were at least three other aircraft operating in the airport environment also making their respective radio reports. Many of these calls were 'stepped-on'. My student urgently called my attention to the impending collision hazard. He had to lean back so that I could see the issue. Immediately I executed a climb and the other plane passed right below us. It was very close. The other plane then asked if we had seen him! I suggested the next time that he enters the pattern it ought to be on the crosswind. The next five seconds of radio communications were nothing but a squeal. This is not the first time that I have experienced the mid field cross-over problem. Intuitively pilots think that they are safe crossing mid field because they are avoiding landing and departing traffic. This is not true. There are lots of aircraft that can be at pattern altitude on take-off by mid field. There are also go-arounds and missed approaches to be considered.Moreover; a pilot doing a mid-field cross-over is looking in the direction in which they intend to turn - they don't see established downwind traffic. As downwind traffic; it not expected to have other traffic effectively truncate the pattern and cut in half way. This generally not the way I am looking for traffic to enter a busy pattern.I would specifically write into the AIM not to do this maneuver. If one needs to enter the pattern from the opposite side; do so above pattern altitude (I suggest 2000 AGL rather than the 1500 AGL as quoted in the Airplane Flying Handbook because would also avoid multi-engine pattern) teardrop or enter the pattern on the crosswind. A crosswind entry allows the pilot to see the whole airport and sequence appropriately.Significantly; pilots are not looking out the window anymore and have been using ADS-B in lieu of their eyes. Lots of planes have no ADS-B. And; ADS-B has a numerous failure modes. The convenience of ADS-B has led [to] laziness and a significant increase in risk versus the initial claims that it would improve safely in the air. It needs to be strongly reemphasized to see and avoid - ADS-B is not an acceptable substitute! Birds don't have ADS-B and they can take you down as readily as another plane. Ask Captain about that one!