No, but it changes whether the original is in the game or not.
My point stands. A piece of recorded music has two things to consider. The first is the rights to the actual master recording, which typically belongs to the label. The second is the writing/publishing rights, which can belong to anyone from the original songwriter to a representative or even the label depending on the contract between artist and label.
The fact that the original itself isn't in the game only gets around the master record copyright. The composition itself still needs to be cleared. If you check the chain of quotes, the original query was about potential reasons it's not on the soundtrack, and what I said applies regardless of if it's the original or a 'cover' sung by in-game characters.
That's the main gist of it which applies to US copyright law. There's some differences in detail in how Japanese labels typically handle their rights, but not anything significantly different in this case.
Incidentally, under US copyright law, a cover could be done under
compulsory license, which still does cost a bit, but circumvents the whole 'let's bargain till everyone's blue in the face' stage. It's still a moot point, though, as Type-0 is developed in Japan. As to why it's in TFF:CC and not on the soundtrack, who knows. ¯\_(ツ
_/¯ I can speculate that the contract was flexible regarding games, but not soundtrack releases. (since labels are way more familiar with music releases than games, and if they smell $$$, they're out for blood)
tl;dr the music industry today is a big mess, especially when pop music gets involved.