Maybe Shueisha has the rights/joint rights to the songs?
Or maybe they just paid the record label. Not sure how much that would cost though.
I looked on some sites that are supposed to tell you the owner of songs and checked Dragon Soul and Cha La Head Cha La and it mostly just gave some Japanese people. One (repertoire.bmi.com) also had for both songs Fujipacific Music, Inc. as publisher, which according to wikipedia is the film music subsection of Fuji TV Group (these guys also broadcasted Pride MMA, until its Yakuza links got made public by journalists and Fuji cancelled their contract, marking the beginning of the end) which is part of Fujisankei Communications Group, which is the biggest media company in Japan. Weirdly I didn't see Hironobu Kageyama listed for Cha La Head Cha La. The Latin American Cha La Head Cha La has Fujipacific and also Toei Animation listed as publishers. There's also some random guy who I think is from Morocco who has Dragon Soul, Cha La and a bunch of other anime songs but nobody else listed on his versions. Idk how complete or accurate the info on the site is though and especially for in Japan.
You could try uploading Dragon Soul or the Saint Seiya one to youtube and see who claims it. Use a Japanese VPN too, maybe that makes a difference.