Once you skillbreak hashirama his passive ceases to be a removable buff and becomes a not removable buff (aka aura) as you pointed out. But exactly like every aura it is in effect as long as the ninja that emanates it is alive. If skillbroken hashi dies the aura effect is not anymore active. Just to explain better: if you fight a blitz team that is not able to remove buffs (like the old darui, wb asuma, iruka, mb blitz) and hashirama dies in round 1 if is not skillbroken then your team continues to have the buff for the following 4 rounds, if is skillbroken you lose it in round 1. You can try by yourself if is really so or not. Make a duel against somebody and use shark bomb in round 1, then wait for him to kill hashi skillbroken and use shark bomb again in round 2 and you'll see that works like i said because your shark bomb will land 20-30% less damage.

Did you really think that a single skillbreak made hashirama passive better than madara passive?