I didn't say most players won't avoid her, I said it's possible, especially if she can surprise her opponents.
here are her numbers
up to 45% max hp
if one E lands, you can then summon 2 tentacles against one champ using ult
350 from ult
W, auto (or inverse as it has been a while since I have been able to play), 290 + 8% of target's max hp
2 tentacle attacks possible at twice normals swing speed, 795 if both tentacles hit
Q for 397
this is the 2nd worst possible setup for an experienced illaoi and the total is 1542 + 8% max hp(2000 = 160) + 45% of all damage done to the spirit (which if the target does not run exclusively can be the same done to the target), this comes to 1542 * 1.45 = 2235
potentially 2395 damage from a situation she was not designed for, now lets see what happens when she hits all 5 opponents
up to 45% max hp
350 from ult
auto attacks for 290 + 8% of target's hp
6 tentacles (potential for 7 if one is already spawned) for 2382 or 2779 if there are 7
Q for 397
this comes to 3579 to 3976 + up to 45% max hp
by the way, this is ONE...I REPEAT, ONE ROTATION
you can add in truckloads more with her W being on low cooldown, Q coming off cooldown, sunfire cape, auto attacks thrown in with iceborne gauntlet, dead man's plate, additional tentacles spawned by the player becoming a vessel, sterak's gauge raising her base AD, even titanic hydra if you want
by adding in those, you come to 100 from dead man's plate, 344 from sunfire cape, a 25% increase to base AD from sterak's that can increase to 50% if she takes at least 400 damage(i'll factor that in when finished with the rest), all base AD from icebourne ( so 181 to 217), titanic hydra adding at least 160 + 10% of illaoi's max health)
not to mention the stupidly large bonus she gets from her W being off cooldown
in that 8 seconds, if handled perfectly and not stunlocked, she can do up to depending upon whether you get cooldown reduction boots or not as well as other factors
that's 12,275 damage to a single target and about 1000 to 2000 less to other opponents without the keystone mastery and without factoring in sterak's completely(it would actually be much higher if I added its number into everything but I chose to just use it in the auto attacks) and without any pure AD items
I'd say a minimal value of 2395 damage on a single target in a fairly inexperienced player's hands is pretty good and if her opponents make the wrong mistake, a flash ult engage from the bush can completely end a teamfight or baron push or dragon...etc etc
there are variables depending upon which tentacles strike the other team members, but she has the highest base damage capability in the game
.
is almost as frustrating, but I think that has more to do with the item than him. There is no counterplay involved when he completes that item.
and 
are very dependent on the situation in my opinion. I would never ban them since I don't feel like their pick rate is high enough. I've seen more than my fair share fall flat on their faces in ranked as well. However when one does get fed it feels hopeless, especially if you don't have many escapes yourself.
for the bottom lane
for mid lane
because he has too many tools. When riot reworks 
have (in comparison to 
)
versus 
