Read each and every quote in that video.
This video gives crystal clear evidence that its the entire room which binds you, your soul is registered the moment you step there...and the only way to proceed is playing the game to its completion. Either die or win are the only two options to escape it. That or you are stuck there permanantly with thumb up your ass.
It isn't the dice that binds you but the room as a whole.
Nero is forced to play this two times in entire game, M06 and M19.
And technically its not Nero but all the demons in it are bound to it too...each slot on the board
creates a dimensions for Nero to fight in and it
recreates the entire battles from the past. Dead Bosses like Berial, Echidna etc are braught back to life just to impede Nero's progress, even they are affected. Entire atmosphere is recreated , from the time of the day to the specific locations intact as they were in the past.
Anyone who has experience playing dice based board games( almost everyone has played of course, if you haven't well shoo shoo then go play one) you know how the deal is....
Lets take Snake and Ladders as example....
The only way to make positive progress forwards is to toss the dice and move accordingly, whether that brings you misfortune or fortune comes later...
If you refuse to play the dice then your familiar on the board remains stagnant on the slot. The dice is just an instrument of progression....
If you have ever had a salty bratty little sibling whom you may have played the game with(speaking from experience here


), and they throw or destroy or hide the dice then the game cannot continue, you are at an impasse....your avatar is stuck there....then you have to spend hrs convincing your sibling to give back the dice, or get a new one or go search for it yourself.....and only then you can continue playing.
But imagine having a player who registers his avatar on the board but gets rid of the dice and proceeds to win the game by just moving the avatar to its win position and get off scot free without consequences.
You know who does that in DMC4...yeah thats pretty blatant fate manip resistance.