The TMHC prototype's "2SHELL_L.VAG" and "2SHELL_R.VAG" files contain a string called "ShellKidRock13R" in their headers. So I checked out that artist's discography on Wikipedia, focusing on the 13th tracks of his pre-2004 albums...

Turned out to be "Drunk In The Morning" from an album called "Cocky". TMHC uses a looping snippet of that track, from roughly 3:50 to 5:00 in the following video:



You may have noticed that in the overview video you linked to, several people identified other tracks from various artists in the comments section. TMHC's versions of those tracks are also just snippets from the full versions (usually focusing on instrumental parts).

It's extremely unlikely that any of these tracks would've made it to the final game. At best, I suppose it's possible that maybe they could've been used as inspiration for the kind of atmosphere Incognito would've wanted to establish in the game's final soundtrack.