When I supp with Illaoi I play her as a poke champ.
She can play a poke support, and she's not significantly stronger than other poke support. However, her all-in strength is countered and reduced much easier than the all-in of other poke supports is (I can kite out Illaoi and kill tentacles to make her unable to all-in me, but I can't do the same to an angry Karma). If all I want is poke, I can do better. If I want poke with other mechanics added on, I can do better.
You also have a few other problems related to things she can't do. First, she can't roam. She has no ability to convert a kill on the midlaner or jungler by roaming on them. She'd need CC to do that, and her E isn't good enough. Supports are expected to roam, and not being able to do so is a major disadvantage. That might be excusable if she could get kills in lane, but she can't. She can poke, she can harass, and she might be able to force the enemy to back (not better than other poke supports can of course, but not worse either). However, unless they choose to stick around, she can't actually finalize kills, again because of the lack of CC. This is a weakness of Illaoi anywhere you play her, but it's a damning weakness in the support role. I might have actually lose a lane against an Illaoi once if she could have taken her good trades and actually gotten a kill on me from them, but she just can't. If I'm playing Zyra I can root them and burst them and finish them off. Karma has burst and her root. Sona has an ult. Illaoi has her Q, and then some melee stuff that she can't use on a cautious opponent. That is unacceptable.
Basically, she works only when her opponents don't know how to play. She works when people don't respect how much power she has in melee range, and she works when people don't abuse how much power she lacks outside of it. That can kind of work in top lane where she doesn't give enemies a choice and where she can mass up enough of a gold and level advantage to shove herself into teamfights and succeed, but it just fails more often than not bot lane.
tbh her damage base is high enough to still be impactful in a teamfight.
Assuming that she doesn't get blown up from being behind on levels and gold, which are huge hurdles for an Illaoi trying to be useful. Even if things go ideally for her, her base damages are not that amazing where she's doing something I couldn't do as a mage support with stronger poke and better utility. I can play Zyra and win teamfights more reliably and without a gold advantage.
she has a hard time playin when behind
That's putting it mildly. I'd personally go with "worse than any true support when behind".
If Illaoi support is working for you, then enjoy it, but I can tell you the reasons why high level players don't respect it, and why it would take a massive amount of practice with Illaoi support to make it succeed against equally-skilled players who know how to play against her.