If an inexperienced MS Sasuke can put a perfect jin in a genjutsu for 2-3 seconds then imagine what a Sharinnegan can do. 3 tomo Sasuke can suppress Kurama's powers and sharinnegan solo'd 9 bijuu at the same time.
Sasuke can easily place Naruto and his 9 pals in a genjutsu at the same time. He can just hop into Nardo's mind and solo the tailed beasts with genjutsu.
Plus, more bijuu =\= faster genjustu counter. The speed at which the genjustu is broken is the speed at which your fastest partner can disrupt your chakra.
Analogy: Imagine that I am caught in a genjutsu and I have 9 friends to break me free. The first friend to disrupt my chakra is the one who breaks the genjutsu. The 8 other friends in this scenario are meaningless because the genjustu was broken by the fastest friend.
Although the way to disrupt genjutsu is to overpower the caster's chakra with your own, and since there are 9 chakras inside Naruto that would mean that the genjutsu would be interrupted quicker since Sasuke's chakra would be overpowered easier thus meaning Naruto would be freed faster.
So judging by how genjutsu is supposed to be broken, lets take in your analogy, what if all 9 friends were helping to break the genjutsu? Since there are multiple people the genjutsu has a chance of being broken at a faster rate.
Now lets take in another analogy, lets say that 9 race car drivers are racing against 1 biker.
Lets say all the race cars and the bike are all at mach 30.
But what if the race car drivers combine their cars together, it would be improved and thus faster so instead of being mach 30, the race car drivers combined car would be mach 180 vs the biker who's speed is mach 30. Now since the race drivers car speed is faster, wouldn't that mean the car would get to the finish line faster than it would if it was still at mach 30?