Not sure to what extent this could be considered an answer to my own question, but it seems this song is one of the many unreleased tracks Darker Than Black has. I have found that many people in Youtube comments lament about this regarding many other DTB songs.
Here is Shout Baby on 緑黄色社会's channel: https://www.youtube.com/watch?v=CMvep1V6rZQ
and you can hear it is indeed the same song as the MHA S4 ED2: https://www.youtube.com/watch?v=Ei0LbNH02-Y (Funimation channel)
始まりの歌 is also on 緑黄色社会's channel here to confirm it's the same artist: https://www.youtube.com/watch?v=cjf9c6TwH6o
(始まりの歌 pretty much translates ...