An instructor pilot a reported they performed a go-around from 15 feet when they observed another aircraft taxiing on to the runway.

Narrative

A seemingly (unsure) antique or rather older looking aircraft; tail dragger; conventional tail; low wing; was ADS-B negative and NOT making radio calls. Listening to the CTAF before coming in to land; other aircraft had issues maintaining separation with this traffic. My student and I flew over the field and did a right teardrop entry to midfield left downwind Runway XX behind another DA40 at ZZZ. Only one other known aircraft was in the pattern; a piper cub. We enter the pattern and do a standard approach to landing. On final and then short final; no traffic is seen on the runway. Almost 15 feet above the runway; the aircraft described; comes into sight right in front of us. We were near the round out and the other aircraft appeared to be trying to takeoff. I immediately perform a go-around; sidestep to the west; climb to a safe altitude and immediately leave the pattern to the west for fear this aircraft climbs up underneath us. I believe they may have taken the runway as we were right above the numbers. I berate the pilot and asked why he wasn't making radio calls. We made radio calls at all stages of the pattern; on final and short final. Our radio frequency was correct and our radio was working as I asked another pilot in the pattern if they knew who the other pilot was. I find it hard to believe this other pilot even had their radio on at all or even had one in the aircraft. They had no reason to be attempting to operate that aircraft. After our go-around we climb to 500 ft over pattern altitude to the west fly over the field and re-enter the left downwind for XX. We do a standard approach to landing and land without further incident. After taxiing back and putting our airplane away; this other aircraft is nowhere to be seen. I presume they aborted takeoff and put their airplane away as fast as they could. A safety report was issued by myself and my student.