You can't really tell from the cropped videos (well, maybe you can by keeping an eye on the life total), but hunting stages are really tedious to speedrun. One of the pieces has 45 possible locations, one has 36, and one has 27. From each of those sets, I only accept the one location that's fastest to reach from the start, so I keep dying until I get each piece where I want it. In SA1, emerald pieces were treated as checkpoints in both the time and the positional sense. In SA2, the positional aspect of the checkpoint is removed, so you return to the single start location all the time. But pieces as "time checkpoints" remain intact. A good place to observe this is in Pumpkin Hill.
I get the first piece at 4.15, then die shortly afterwards at 4.44. I end up returning to the starting point, with a timer of 4.15 instead of 4.44.
The effect is more pronounced on the second segment. The piece is obtained at 8.45, but I don't die until 25.75 (in order to glide to the outer edge so I can set up the rising start, an important timesaver in Pumpkin Hill). But the piece is the checkpoint, not the kill time, so the clock reverts back to 8.45 for the final segment. The extra 17 seconds don't count against me at all, and I return to start (albeit not for long, as I get shot upwards).
When you have one piece location, the game looks for nearby locations for subsequent pieces and issues a lockout on those locations, making sure never to put the piece there. This is to prevent degenerate layouts where all the pieces are in the same spot. The game doesn't want hunting stages to be that easy. However, since the lockouts only go in one direction, you can subvert this by getting the pieces in reverse order: 3, then 2, then 1. I'm not sure exactly what the lockout radius is, but I've done some testing with a possible new strategy in Meteor Herd that involves getting two pieces on the same life. In terms of probability, the two should coexist once every 1215 tries, but I've spent over 3000 with no success.
Basically, read the guide as pointed out in the previous post.