She's nothing but a Base Stat monster. If you nerf her, she'll be useless, so you have to keep her base stats high enough that she'll absolutely crush any living thing that comes near her.
Basically juggernauts in general. High damage and highly tanky (in various ways). Traditionally you need to kite them and you should respect their strengths given to them.
Her Q deals a shit-ton of damage and heals her.
While not a complete trend for juggernauts, it isnt uncommon. Darius, Nasus, and Yorick all deal a lot of damage on their Q's and heal for some amount.
Her W deals a shit-ton of damage and makes her Q and heal a shit-ton more times.
Her W deals bonus damage as max-target-hp damage. If she has no nearby tentacles, she still needs some form of reliable damage and it comes mostly from her W since it has a shorter CD than her Q and doesnt root her. Her W doesnt affect her Q at all. you're thinking about her passive spawning nearby tentacles. which is what you really should be avoiding or killing when possible.
Her E either amplifies that damage a shit-ton or forces you out of lane for an unhealthy ten seconds.
Her E has good range but is kinda slow. Its not hard to avoid and gets blocked by various units. If it's amplifying damage, then you're simply too close to her to begin with. otherwise, you walk away alive. And it doesnt force you out of lane, it just makes it a minigame of sidestep a slow tentacle every 5 secs. not hard. walk perpendicular to the trajectory. A tentacle also wont swing more than once at a vessel, so if you're near a tentacle that spawned and swung already, you're safe to stand by it until the debuff wears off and then you can kill it without another spawning. Also, it used to be a FULL MINUTE and spawned a tentacle every 10secs for 60secs. having a tentacle spawn every 5-3 secs for 10-15secs is must better for anyone playing against her in regards to "leaving lane"
Her R puts her W on a infinitesimally small cooldown, allowing her to use her Q at least ten times in four seconds... if she's not dumb enough to let you walk away.
10 times in 4 secs? I never knew 2sec went into 8secs 10 times. Her R changes the W CD to a flat 2 secs. CD reduction doesnt make it any lower than 2sec during her ult. Outside the ult duration, the W CD is usually 2.2-2.4secs depending on if you have 40-45% CD. Her ult does however, reduce the Q cast time to 0.5sec from 1sec since it is essentially the same as a wall-tentacle swinging.
In regards to the last part "if she's not dumb enough to let you walk away". If you have a champ with any form of "higher" mobility or a strong "peel" effect, illaoi can't do much about it. Nasus W is strong against her if they're not running stormraiders. Renekton has his E. Gnar has his E, nautilus has enough CC to self-peel and his Q has long range terrain-hook. Wukong (while not the best pick against illaoi imo) can ult and E away. Even an illaoi with Cleaver, IBG and flash won't catch up to some of them.
Since her base stats are so high, she's allowed to go full tank after one item, which when combined with her many, many %missing health heals, does make her way harder to kill than mostly any other champ in the game.
Darius goes cleaver and then tank. Garen goes cleaver and tank usually. Yorick and Nasus both go Sheen-Item and tank. Trundle and Olaf usually go Botrk/hydra and then full tank. Volibear and mundo are designed to scale mostly off health and generally dont need a damage item. Even mordekaiser can go hextech gunblade into full tank (especially since the rylai's nerfs). This is just how juggernauts are designed.
The only counter-argument she gets is "Get out of the way xDDDDD." That's exactly the problem. She's a champ that's designed for people to avoid. That means either multiple people have to focus her down, leaving the other 4 teammates to do what they please, or people ignore her completely, giving her a free lane.
Illaoi is designed to pick a spot and keep it (passive is the example). She claims that area as hers and not yours. If you want that area, you are tested. If you win the test, she is forced to back off and you can hold that spot for a while. If you fail, you will be wounded, or worst. Then she will attempt to claim more of your land. This is what illaoi does, she picks a spot, claims it, and challenges you to take it back. Its not actually about "avoiding her", its about how do you keep her from taking more. Illaoi loves head-first dives usually and this is where people fail the test even before it begin.
In addition, juggernauts tend to be able to fight multiple opponents when ahead. if a darius or nasus (her equivalents in my opinion) are doing well in lane, you generally send 2-3 people to deal with them appropriately. But that doesnt mean you can go in all brute-like. A good juggernaut is the equivalent to a strong splitpusher (or both). They become strong threats that must be dealt with by multiple opponents, otherwise, they will continue their rule of the land.
She offers nothing to her team except damage. She has no CC except one slow that'll only proc after the fight is over. If you reduce her damage, she's nothing.
She may not have any CC like a darius or a trundle, but her damage is a strong enough deterrent and her tentacles make you feel outnumbered. Illaoi's engage is generally slow unless she gets her spirit pull on enemies, otherwise, she will soak up poke damage or wait in the mid-line and act as a damage-based disengage.
She has been in an incredibly unhealthy state for as long as she's been released
Unhealthy or unbalanced? I consider Fiora, and gnar unhealthy but i know they're balanced for the most part.
She's not fun to play. She's not fun to play against.
She is plenty fun to play, and i've also played against her with various champs (sion, nautilus, Nasus, veigar, renekton, mundo, chogath, maokai, and pantheon) and while i consider it a challenge, i wouldnt call it unfun.
She's not fun to have on your team. She's just a gigantic circle in which everything that steps inside of dies, and that will never, ever be fair to anyone who has to put up with her.
This is essentially the definition of a juggernaut who doing well. You dont walk up to a darius who isnt behind, you dont walk up to a nasus with Q stacks. and you dont walk up to a Yorick with his ult off-CD.
Juggernauts are "you get close to me, i will do damage". This is why their mobility and any forms of CC are kept an eye on when released.