B767-300 Check Airman reported on final approach the pilot flying and on an OE initiated a go-around and reporter took control of aircraft. On the second approach the aircraft was again unstable and the pilot flying did another go-around. The third approach and landing were successful.

Narrative

I was conducting OE with a new hire. We had briefed the go-around and talked about it again briefly on downwind. The FO (First Officer) did the RNAV XX. The missed approach altitude is 2;000 ft.; and we were in a light airplane. The approach was excellent but he floated the landing so I called go-around and initially everything was going great. At 1;000 ft.; he called FLCH set speed 190. It was a little disarming because until this point everything had gone so well. I looked down to switch us to Departure but before I made the call I looked up and saw we were at 1;500 ft.; and climbing at an extreme rate with rapidly increasing airspeed. I immediately took the aircraft went to flight idle and executed a smooth pushover. We peaked about 300 ft.; high. I established a slight descent; called for FLCH; flaps 5; climb thrust and called for the center Autopilot. I am keenly aware of the danger in handing the aircraft to the Autopilot out of trim but I must have. ZZZ Approach gave us 3;000 ft.; so my eyes went up to the MCP (Mode Control Panel) to set that and initiate the climb. By the time my eyes got back to the instruments we were 300 ft.; low and descending. I intervened again; the low point being almost 500 ft.; low; ZZZ advised us we were descending along with a clearance to climb. The rest of the missed approach went without incident.I opted to fly the second approach. The FO quickly setup the RNAV XX and I briefed it. The second missed approach was completely my fault and I did something I have never done before. The final approach fix is at 1;700 ft.; so we were configuring and slowing as the VNAV initiated the descent. At about 1;500 ft.; I rolled up the altitude to set the missed approach. I'm not sure what caused the aircraft to go into VNAV SPD. I may have rolled up the altitude too slowly and the VNAV caught and went to VNAV SPD or maybe it was failing to re-cruise the box. I quickly tried to reselect VNAV thinking it would grab the glide path only to look down and see that we were not only in VNAV SPD but the glidepath was completely absent from the scale. I immediately went around without incident. During the go-around; ZZZ advised us we were low. Thankfully we had ample fuel due to unused contingency fuel and this was briefed on all 3 approaches.As for the first missed approach; I was disarmed by the fact that it started so well and I assumed he had it under control. When he called for FLCH; set speed" my mind transitioned to; the [problem] is over; I can start getting caught up on other things.As for how we got low; I believe it was a confluence of a little adrenaline causing me to not notice the aircraft was out of trim coupled with a massive thrust change and bringing the flaps up to 5. It was too much for the Autopilot. As for the second approach; I'm not sure if it was setting the missed to slowly or failure to re-cruise that caused the box to go to VNAV SPD. We had VNAV PATH heading into the final approach fix and it was doing a great job descending on the approach until that point. As for how I got low on that approach; it's not hard to see how the aircraft would have descended below the path if the thrust never came back up from the slowing to final speed When the VNAV logic flipped and it initially went unnoticed because large speed and pitch changes are very normal in that phase of flight. Also; my eyes were away from the flight instruments during this critical transition verifying the speed brake was armed and gear was down in addition to resetting the MCP.While I had rested optimally prior to starting my day; it was XA:30 and the second leg of the day so I think a little fatigue played a part here as well.I probably should have emphasized and pushed a new FO to visualize exactly how the missed would go; to include the need to proactively reduce thrust with a light aircraft and a low alt cap. That will be better emphasized in the future. This incident also highlights how a little adrenaline can influence my perception of things and cause me to miss things that I normally would be keenly aware of. I should have taken a deep breath before re-automating not after. It could have waited another 2 seconds in this scenario.I am wondering if it might be best to have the NFP (non-flying pilot) select the missed approach altitude so that the FP (flying pilot) can be completely on the flight instruments instead of having our attention divided at a time when we are low to the ground and headed down. Regardless of the cause and timing; I certainly would have noticed it sooner if my eyes were not on the MCP. I will also stop flying like a normal line pilot with new FO's. I need to be checklist complete before I hit the final fix; if at all possible.This series of events highlights the need for better FMA (flight mode annunciator) awareness; detailed briefing of cadence and the delicate balance of aviate; navigate and communicate. 3 times in 10 minutes-a quick glance away at the wrong time allowed deviations to be exacerbated."