No, they did not accidentally initiate a go-around.
It's semantics. From your link:
The pilot flying (PF) selected an autopilot (A/P) mode (flight level change speed [FLCH SPD]) that instead resulted in the autoflight system initiating a climb because the airplane was below the selected altitude
The autopilot had previously been configured with the go-around altitude set, so when the pilot accidentally selected the incorrect mode ("flight level change speed"), the autopilot advanced the throttles in order to climb to the go-around altitude. It was configured this way so that in the event of a go-around, they'd simply select this mode and the plane would automatically climb to the appropriate altitude. So it was set up for the possibility of a go-around, and the pilot accidentally engaged it.
Upon immediately realizing his error, he manually pulled the throttles back to idle, and turned the autopilot off. This unusual sequence of commands is what disabled the autothrottles.
What really happened was that they were too high above the glide slope when they were 5 nm out from the runway. In trying to descend, they tried to slow down by pulling the throttle levers back. This action overrode the Autothrottle which had been controlling speed, and it went into HOLD mode, and stayed inactive. This was not noticed by the aircrew.
That's not exactly correct. Pulling the throttles back on its own would not have disconnected the autopilot. It was the odd combination of selecting the FLCH SPD mode, then turning it off, then manually overriding the throttles, that resulted in the unnoticed autothrottle disengagement.
At 100 ft, they gave up and did a go-around. The Autothrottle probably responded properly and advanced up
No, the autothrottles weren't engaged - that's the whole point. The pilots had complete control of the airplane. In its final configuration, the computer wasn't controlling anything. The pilots were in full manual control of the aircraft. They manually advanced the throttles to go-around power, but as you noted, big engines like that have nontrivial lag time, and it was too late for them to begin producing meaningful thrust to avoid a crash.
If the Autopilot was off, then of course the corresponding display would be off.
No, I don't mean his autopilot display was off - I mean his primary flight display (PFD) was off. The big, tablet-shaped panel front-and-center in the middle of the dashboard in front of each pilot. I don't know why any pilot would ever turn that off, as it conveys all sorts of useful information about heading, speed, attitude, altitude, weather, etc., but that's what the report said.