rinnegan is the strongest eyes you can have
From what we know of both the rinnegan and the EMS i would say the EMS is stronger, but thats just me...
I'd think that the EMS is stronger than the Rinnegan for the main reason that you could probably just catch one of them in Tsukuyomi and catch all of them at the same time as they all use the same vision, I believe Jiraya managed it but I don't remember that well enough.
On a somewhat related note I'd rather have the Byakugan over the EMS for a reason I haven't seen anyone explicitly mention. You can watch the exact accumulation of chakra and prepare accordingly. Its much more than just countering Taijutsu to me, it also seems to be one of the most effective way of counter Nin- and Gen- jutsus.
From there I'd rather have the Rinnegan over the Byakugan because we all know that the Byakugan's defense isn't perfect, as demonstrated Kidomaru and Neji. And as the old adage goes "The best defense is a good offense."
the only way to beat the byakugan by using its weakness is alot of distance, if you fighting close combat, that one degree is negligible, now if your over a 100 feet away like kidomaru, then you can take advantage of the weakness. and im not sure but i think you cant put a byakugan user into a genjutsu.
What if it's a sound type genjutsu .
Or you could just always do what Shikamaru did and cause a brief moment of intense pain such as breaking a finger.